diff --git a/_images/domaindecomposition.png b/_images/domaindecomposition.png new file mode 100644 index 00000000..f0059421 Binary files /dev/null and b/_images/domaindecomposition.png differ diff --git a/_sources/sobolevspaces/Traces.ipynb b/_sources/sobolevspaces/Traces.ipynb index feebc5db..d9fba33e 100644 --- a/_sources/sobolevspaces/Traces.ipynb +++ b/_sources/sobolevspaces/Traces.ipynb @@ -253,6 +253,8 @@ "The interfaces are $\\gamma_{ij} = \\overline \\Omega_i \\cap \\overline \\Omega_j$.\n", "The outer normal vector of $\\Omega_i$ is $n_i$.\n", "\n", + "\"Alternative\n", + "\n", "> **Theorem:**\n", "> Let $u \\in L_2(\\Omega)$ such that\n", "> * $u_i := u|_{\\Omega_i}$ is in $H^1(\\Omega_i)$, and $g_i = \\nabla u_i$ is its\n", diff --git a/searchindex.js b/searchindex.js index 6bc2a2f2..e7613cbc 100644 --- a/searchindex.js +++ b/searchindex.js @@ -1 +1 @@ -Search.setIndex({"docnames": ["DG/Nitsche", "DG/elliptic", "DG/elliptic_stdDG", "DG/fourthorder", "DG/instationary", 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Nitsche\u2019s Method for boundary and interface conditions", "40. Hybrid DG for elliptic equations", "DG - Methods for elliptic problems", "42. Fourth Order Equation", "37. Instationary Transport Equation", "41. Splitting Methods for the time-dependent convection diffusion equation", "36. Stationary Transport Equation", "43. H(div)-conforming Stokes", "22. Finite element error analysis", "20. Finite Element Method", "24. Implementation of Finite Elements", "23. Non-conforming Finite Element Methods", "21. Finite element system assembling", "84. NGSolve - PETSc interface", "79. Distributed Meshes and Spaces", "<no title>", "<no title>", "<no title>", "<no title>", "78. Introduction to MPI with mpi4py", "82. Iteration methods in parallel", "85. Solving Stokes in parallel", "83. Using PETSc", "81. Consistent and Distributed Vectors", "This repository contains an interactive introduction to the Finite Element Method", "7. Basic properties", "10. 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Parameter Dependent Problems", "32. Mixed Methods for second order equations", "30. Stokes Equation", "Non-linear dynamics", "\\(\\DeclareMathOperator{\\opdiv}{div}\\)\nHellinger Reissner mixed formulation", "\\(\\DeclareMathOperator{\\opdiv}{div}\\)\n\\(\\DeclareMathOperator{\\opcurl}{curl}\\)\n\\(\\DeclareMathOperator{\\eps}{\\varepsilon}\\)\nReduced symmetry methods", "Tangential displacement normal normal stress continuous finite elements", "62. Multigrid and Multilevel Methods", "64. Analysis of the Multigrid Iteration", "63. Analysis of the multi-level preconditioner", "65. Multi-level Extension", "The Hellan Herrmann Johnson Method for Kirchhoff plates", "Relationship between HHJ and TDNNS", "Reissner Mindlin Plates", "Nonlinear Shells", "3D-TDNNS Elasticity and Reissner Mindlin Plate elements", "Preamble", "2. Boundary Conditions", "5. 3D Solid Mechanics", "6. Exercises", "1. Solving the Poisson Equation", "4. Iterative Solvers", "3. Variable Coefficients", "29. hp - Finite Elements", "67. The Bramble-Pasciak Transformation", "68. A Small Number of Constraints", "70. Parameter Dependent Problems", "66. Structure of Saddle-point Problems", "<no title>", "47. Finite Element Error Analysis", "48. Error Analysis in \\(L_2 \\times H^1\\)", "46. Finite Elements in \\(H(\\operatorname{div})\\)", "44. Application of the abstract theory", "45. The function space \\(H(\\operatorname{div})\\)", "49. Hybridization Techniques", "Friedrichs\u2019 Inequality", "<no title>", "<no title>", "Friedrichs\u2019 Inequality", "The Poincar\u00e9 inequality", "The Trace Inequality", "13. Generalized derivatives", "14. Sobolev spaces", "15. Trace theorems and their applications", "16. Equivalent norms on \\(H^1\\) and on sub-spaces", "19. Exercises", "18. Experiments with norms", "17. The weak formulation of the Poisson equation", "56. Additive Schwarz Methods", "57. Some Examples of ASM preconditioners", "61. 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71, 75, 79, 81, 85, 86, 95, 97, 98, 99, 101, 102, 106, 108, 110, 111, 112, 118], "navier": [5, 11, 56], "stoke": [5, 11, 43, 44, 81, 98], "easili": [5, 23, 32, 39, 45, 46, 47, 62, 63, 77, 109], "also": [5, 8, 9, 11, 14, 19, 25, 26, 27, 28, 29, 31, 32, 33, 34, 38, 39, 45, 46, 47, 48, 49, 51, 53, 56, 61, 63, 70, 74, 77, 80, 81, 86, 95, 96, 97, 100, 101, 102, 103, 108, 109], "system": [5, 25, 29, 32, 35, 36, 39, 40, 44, 45, 47, 48, 49, 50, 52, 54, 55, 56, 59, 65, 74, 75, 78, 79, 81, 86, 88, 108, 109, 110, 111], "difficult": [5, 103], "implicit": [5, 112], "explizit": 5, "treatment": [5, 72], "would": [5, 33, 52], "lead": [5, 9, 11, 26, 27, 37, 46, 50, 51, 61, 65, 77, 80, 81, 85, 86, 88, 94, 96, 102, 103, 105, 108, 109], "sever": [5, 11, 38, 56, 62, 105], "step": [5, 8, 11, 12, 20, 32, 33, 34, 38, 43, 45, 46, 48, 49, 61, 62, 68, 77, 79, 81, 102, 103, 105, 106, 108, 112], "fast": [5, 44, 48, 97, 100, 109], "becom": [5, 11, 38, 49, 86, 100, 115], "begin": [5, 6, 8, 9, 11, 12, 23, 26, 27, 29, 31, 32, 33, 34, 35, 36, 38, 39, 45, 46, 48, 49, 50, 51, 52, 53, 54, 55, 56, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 71, 76, 77, 78, 79, 80, 81, 83, 84, 86, 87, 88, 89, 94, 95, 96, 97, 98, 99, 101, 102, 103, 106, 108, 109, 110, 111, 112, 118], "arrai": [5, 6, 11, 22, 23, 36, 38, 50, 51, 53, 54, 55, 56, 58, 59, 60, 65, 66, 67, 71, 74, 78, 79, 81, 83, 86, 88, 95, 97, 99, 102, 108, 109, 118], "cc": [5, 23, 36, 59, 71, 79, 81, 88, 108, 118], "end": [5, 6, 8, 9, 11, 12, 15, 23, 26, 27, 29, 31, 32, 33, 34, 35, 36, 38, 39, 45, 46, 48, 49, 50, 51, 52, 53, 54, 55, 56, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 71, 76, 77, 78, 79, 80, 81, 83, 84, 86, 87, 88, 89, 94, 95, 96, 97, 98, 99, 101, 102, 103, 106, 108, 109, 110, 111, 112, 115, 118], "c": [5, 8, 9, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 25, 26, 27, 28, 32, 35, 36, 40, 44, 45, 46, 47, 49, 51, 52, 54, 61, 62, 63, 65, 71, 75, 77, 78, 79, 80, 81, 82, 87, 88, 89, 92, 93, 94, 95, 96, 97, 98, 99, 100, 101, 102, 103, 105, 106, 108, 109, 110], "diff": 5, "conv": [5, 16], "ep": [5, 80, 103], "adiff": 5, "aconv": 5, "mstar": [5, 110, 111, 112], "asvector": [5, 40], "0x10ed11eb0": [], "convu": [5, 16], "keyboardinterrupt": 5, "traceback": [5, 117], "most": [5, 45, 96, 105, 108, 117], "recent": [5, 44, 60, 115, 117], "last": [5, 8, 10, 32, 33, 38, 47, 49, 74, 77, 81, 94, 106, 117], "cell": [5, 10, 19, 44, 74, 107, 117], "line": [5, 8, 9, 29, 30, 33, 45, 48, 84, 117], "15": [5, 14, 20, 36, 37, 46, 61, 75, 78, 80, 81, 82, 88, 107, 115], "13": [5, 14, 20, 36, 37, 46, 61, 75, 78, 80, 81, 82, 88, 107], "14": [5, 14, 20, 36, 37, 46, 61, 75, 78, 80, 81, 82, 88, 107], "16": [5, 20, 36, 37, 46, 61, 65, 74, 75, 78, 80, 81, 82, 85, 88, 100, 105, 107], "17": [5, 20, 36, 37, 46, 60, 61, 65, 74, 75, 78, 80, 81, 82, 88, 107], "higher": [5, 6, 8, 11, 71, 74, 88, 97, 115, 118], "imex": 5, "see": [5, 10, 11, 29, 35, 44, 51, 52, 56, 57, 58, 71, 74, 77, 78, 86, 97, 100, 112], "rung": [5, 43], "kutta": [5, 43], "pde": [5, 23, 26, 29, 38, 97, 102], "uri": 5, "ascher": 5, "steven": 5, "j": [5, 9, 12, 19, 28, 29, 32, 35, 36, 37, 40, 44, 46, 47, 49, 62, 65, 74, 77, 80, 85, 96, 97, 102, 105, 106, 108, 109, 110, 111, 112, 115, 116, 117, 118], "ruuth": 5, "raymond": 5, "spiteri": 5, "numer": [5, 11, 12, 34, 44, 47, 50, 74, 80, 81, 83, 100, 103, 108, 109, 112, 118], "mathemat": [5, 32, 44, 74, 100], "25": [5, 14, 20, 36, 37, 46, 72, 75, 78, 80, 81, 82, 85, 88, 105, 107], "1997": 5, "text": [6, 28, 32, 41, 45, 46, 47, 48, 49, 51, 53, 55, 56, 58, 59, 60, 61, 63, 65, 66, 68, 71, 72, 74, 77, 79, 80, 81, 85, 86, 87, 97, 99, 102, 103, 104, 105, 106, 108, 109, 110, 111, 112], "model": [6, 56, 67, 68, 71, 74, 75, 76, 115], "inject": [6, 28, 29, 97], "concentr": 6, "flow": [6, 26, 81, 109], "exampl": [6, 9, 11, 19, 25, 26, 29, 32, 39, 43, 44, 50, 51, 61, 71, 74, 79, 80, 82, 95, 100, 105, 107, 110, 111], "milk": 6, "coffe": 6, "u_": [6, 23, 29, 32, 34, 40, 51, 62, 63, 74, 89, 92, 94, 101, 103, 112], "inflow": [6, 15, 17, 18], "popular": 6, "Their": 6, "strength": 6, "techniqu": [6, 33, 38, 43, 44, 62, 77, 97, 98, 102, 109], "stabl": [6, 28, 43, 44, 52, 58, 64, 77, 80, 106, 108, 109], "combin": [6, 8, 10, 11, 14, 28, 34, 47, 49, 56, 61, 62, 63, 74, 97, 102, 107, 112, 118], "seen": [6, 47, 51, 61, 62, 80, 88, 96], "extens": [6, 11, 22, 25, 35, 43, 44, 61, 77, 88], "trial": [6, 10, 74, 76], "piecewis": [6, 31, 34, 97, 105, 106], "follow": [6, 8, 10, 11, 29, 31, 32, 33, 34, 35, 38, 39, 44, 45, 49, 51, 52, 61, 62, 63, 64, 71, 77, 80, 84, 85, 88, 94, 95, 96, 97, 98, 100, 102, 103, 104, 105, 106, 108, 118], "wise": [6, 14, 23, 32, 35, 56, 58, 68, 74, 83, 85, 103, 118], "smooth": [6, 8, 9, 12, 31, 39, 58, 61, 62, 87, 95, 96, 97, 101, 103], "evei": 6, "sum_": [6, 8, 9, 11, 12, 23, 31, 32, 33, 34, 36, 38, 39, 40, 45, 47, 49, 60, 63, 64, 74, 77, 84, 87, 88, 96, 97, 98, 99, 101, 102, 103, 108, 110, 118], "subset": [6, 8, 9, 11, 12, 14, 25, 27, 29, 32, 34, 35, 38, 39, 45, 47, 52, 61, 62, 64, 65, 66, 74, 77, 80, 81, 85, 89, 92, 94, 95, 97, 98, 99, 101, 102, 103, 106, 108, 109, 110], "b_n": 6, "outward": 6, "decis": 6, "blow": 6, "up": [6, 8, 11, 13, 14, 23, 26, 28, 29, 47, 49, 56, 61, 64, 77, 80, 94, 95, 96, 98, 106, 108, 109, 110, 118], "cl": [6, 11, 34, 95, 99], "u_t": [6, 60], "outflow": [6, 15, 17, 18, 55], "v_h": [6, 8, 11, 12, 23, 26, 27, 29, 32, 33, 34, 52, 60, 63, 66, 74, 80, 81, 83, 84, 88, 100, 102, 103, 106, 109, 110], "u_h": [6, 8, 11, 12, 26, 27, 29, 31, 32, 33, 34, 52, 60, 63, 74, 80, 83, 84, 88, 100, 102, 106, 109, 110], "all": [6, 8, 9, 10, 14, 19, 23, 25, 27, 28, 29, 30, 31, 32, 34, 35, 37, 44, 47, 49, 50, 51, 52, 61, 63, 64, 67, 70, 71, 72, 74, 80, 87, 96, 97, 98, 99, 101, 102, 103, 105, 106, 107, 108, 109, 110, 112, 118], "6": [6, 13, 20, 22, 36, 37, 38, 42, 45, 46, 53, 61, 68, 72, 74, 75, 76, 77, 78, 79, 80, 81, 82, 85, 88, 100, 105, 107, 108, 109, 112, 115], "28": [6, 20, 36, 37, 46, 74, 75, 78, 80, 81, 82, 107], "hybrid": [7, 43, 44, 52, 65, 70], "lehrenfeld": [7, 70], "sch\u00f6berl": [7, 43, 44, 60, 69, 70, 80, 116, 117], "2016": 7, "bdm": [7, 32, 85], "k": [7, 8, 25, 26, 32, 39, 45, 46, 47, 48, 49, 56, 63, 64, 65, 77, 81, 83, 84, 85, 88, 95, 96, 97, 98, 99, 101, 109], "k_": 7, "splinegeometri": [7, 42, 60, 78, 82, 86], "geo": [7, 35, 36, 37, 42, 60, 68, 72, 76, 78, 79, 80, 82, 86, 105, 115, 118], "addrectangl": [7, 42, 60, 78, 82, 86], "41": [7, 14, 15, 17, 18, 20, 21, 36, 37, 46, 74, 75, 78, 80, 81, 82, 107], "wall": [7, 15, 17, 18, 21, 78, 82], "outlet": [7, 15, 17, 18, 21, 78, 82], "inlet": [7, 15, 17, 18, 21, 78, 82], "addcircl": [7, 42, 78, 82, 86], "leftdomain": [7, 42, 78, 82, 86], "rightdomain": [7, 42, 78, 82, 86], "cyl": [7, 15, 21, 68, 72, 75, 78, 82], "08": [7, 20, 36, 37, 46, 61, 75, 78, 80, 81, 82, 88, 107, 112], "0x105d6ea50": [], "vt": 7, "vf": 7, "tangentialfacetfespac": [7, 16, 17, 18, 113], "q": [7, 9, 21, 25, 32, 51, 52, 53, 54, 56, 60, 76, 77, 78, 81, 82, 97, 98, 118], "nu": [7, 15, 17, 18, 72, 80], "tang": [7, 16, 17, 18, 60, 65, 67, 68], "thesi": [7, 57, 60, 70], "christoph": [7, 70], "page": [7, 45, 47], "71": [7, 36, 37, 46, 75, 80, 82, 107], "invstok": 7, "uin": [7, 15, 17, 18, 21, 78, 82], "re": [7, 32, 34, 46, 48, 49, 61, 79, 110], "vel": 7, "veloc": [7, 15, 56, 57, 81, 111], "pressur": [7, 56, 80, 81], "let": [8, 11, 12, 19, 24, 25, 26, 27, 28, 29, 30, 31, 34, 35, 38, 39, 45, 46, 47, 48, 60, 61, 62, 63, 64, 74, 77, 79, 80, 81, 83, 89, 92, 94, 95, 96, 97, 98, 99, 101, 102, 103, 105, 106, 107, 110], "variat": [8, 28, 43, 44, 51, 58, 65, 71, 73, 74, 76, 79, 80, 81, 83, 84, 88, 98, 101, 102, 111], "problem": [8, 10, 11, 12, 20, 32, 33, 35, 36, 37, 38, 39, 45, 48, 49, 58, 61, 62, 63, 65, 67, 68, 73, 74, 75, 77, 79, 83, 86, 88, 94, 97, 98, 100, 101, 102, 103, 104, 105, 106, 109], "sub": [8, 11, 22, 23, 25, 28, 30, 32, 35, 36, 37, 38, 40, 47, 51, 52, 61, 63, 71, 74, 75, 77, 79, 80, 81, 85, 95, 100, 101, 103, 105, 106, 108, 109, 110], "cea": [8, 11, 62], "": [8, 9, 11, 17, 18, 24, 25, 26, 27, 28, 30, 32, 37, 43, 44, 51, 52, 54, 62, 68, 69, 70, 72, 73, 74, 75, 79, 81, 89, 97, 98, 101, 102, 108, 109, 112, 115, 118], "lemma": [8, 9, 25, 26, 29, 30, 34, 35, 43, 61, 62, 63, 77, 80, 97, 98, 103, 105, 106, 109], "bound": [8, 9, 11, 25, 26, 28, 29, 31, 32, 33, 34, 45, 46, 47, 49, 63, 80, 81, 94, 95, 96, 97, 98, 99, 100, 101, 103, 105, 106], "best": [8, 26, 52, 65, 77, 107], "_v": [8, 11, 25, 26, 27, 28, 29, 31, 33, 34, 51, 52, 73, 97, 98, 101], "leq": [8, 9, 11, 25, 26, 27, 28, 29, 30, 31, 32, 34, 35, 45, 46, 47, 48, 49, 51, 52, 59, 62, 63, 64, 65, 69, 77, 80, 81, 83, 87, 89, 92, 93, 94, 95, 96, 97, 98, 99, 100, 101, 102, 103, 105, 106, 109, 112], "inf_": [8, 11, 26, 28, 29, 30, 35, 39, 46, 52, 60, 63, 65, 77, 80, 97, 98, 102, 103, 105, 106], "constant": [8, 9, 10, 11, 14, 26, 27, 28, 31, 32, 34, 38, 40, 49, 52, 56, 57, 61, 62, 64, 65, 74, 76, 77, 85, 89, 95, 97, 98, 99, 100, 103, 105, 106, 108, 109, 118], "factor": [8, 26, 29, 46, 61, 62, 64, 68, 74, 77, 80, 81, 94, 95, 97, 98, 105, 106, 118], "ratio": 8, "take": [8, 13, 24, 27, 29, 30, 31, 34, 38, 39, 44, 52, 60, 63, 74, 76, 81, 84, 86, 96, 97, 101, 103, 105, 109], "interpol": [8, 9, 11, 16, 34, 44, 52, 63, 64, 71, 72, 73, 74, 80, 83, 106, 115, 118], "i_": [8, 9, 102], "cal": [8, 9, 11, 12, 31, 33, 34, 95, 96], "dimension": [8, 10, 25, 26, 27, 28, 29, 47, 50, 74, 77, 81, 84, 97, 98, 100, 103, 112], "domain": [8, 9, 10, 11, 23, 31, 35, 36, 37, 40, 41, 51, 61, 63, 64, 71, 74, 79, 80, 94, 95, 96, 98, 100, 101, 110, 116, 117], "relat": [8, 12, 27, 45, 48, 62, 76, 98], "affin": [8, 9, 58], "transform": [8, 9, 10, 11, 43, 44, 49, 74, 79, 82, 94, 97, 103], "f_t": [8, 9, 11, 12, 23, 32, 60], "rightarrow": [8, 9, 11, 25, 26, 27, 28, 29, 30, 32, 33, 34, 35, 39, 48, 51, 52, 54, 56, 64, 66, 67, 72, 73, 74, 76, 77, 80, 94, 95, 96, 97, 98, 99, 100, 101, 102, 103, 106, 109, 110, 112], "mathbb": [8, 9, 11, 12, 25, 26, 28, 29, 30, 33, 34, 35, 46, 47, 48, 49, 50, 61, 64, 72, 73, 74, 76, 79, 81, 89, 92, 94, 95, 96, 97, 98, 99, 100, 101, 103, 108, 109, 112, 118], "regular": [8, 9, 10, 31, 32, 37, 51, 60, 62, 63, 74, 80, 81, 88, 95, 101, 102, 106], "Then": [8, 11, 12, 14, 19, 23, 25, 26, 27, 28, 29, 30, 32, 34, 36, 37, 39, 44, 45, 46, 48, 51, 52, 61, 62, 63, 64, 77, 80, 81, 89, 94, 95, 96, 97, 98, 101, 102, 103, 105, 109, 115], "hold": [8, 9, 11, 25, 26, 27, 29, 30, 31, 32, 39, 47, 48, 49, 51, 61, 62, 63, 64, 72, 74, 77, 80, 87, 94, 95, 96, 97, 98, 100, 101, 102, 103, 106, 109], "det": [8, 85, 97, 103, 109, 118], "align": [8, 9, 12, 26, 27, 29, 45, 46, 48, 49, 50, 77, 95, 97, 98, 101], "x_": [8, 11, 47], "i_m": 8, "ldot": [8, 9, 12, 28, 32, 45, 46, 47, 48, 49, 61, 64, 74, 77, 87, 95, 97, 101, 102, 108, 109, 110, 112], "i_1": [8, 39], "j_m": 8, "j_1": 8, "b_": [8, 51, 108], "equat": [8, 11, 12, 26, 29, 31, 32, 33, 34, 36, 38, 39, 46, 50, 52, 53, 59, 63, 71, 72, 75, 76, 77, 79, 81, 86, 88, 97, 98, 102, 108, 115, 116, 117], "preceq": [8, 11, 33, 34, 62, 63, 64, 77, 80, 97, 98, 101, 103, 105], "proof": [8, 11, 25, 26, 27, 29, 30, 32, 34, 35, 52, 62, 63, 64, 73, 77, 80, 81, 84, 85, 87, 89, 95, 96, 97, 98, 101, 102, 103, 105, 106, 109], "chain": [8, 85, 97, 103], "box": [8, 11, 21, 25, 26, 27, 29, 30, 63, 72, 75, 76, 77, 80, 82, 94, 96, 97, 98, 101, 102, 105, 118], "diamet": 8, "h_t": [8, 11, 32, 34], "operatornam": [8, 23, 25, 26, 27, 29, 36, 39, 44, 47, 51, 52, 54, 55, 56, 57, 59, 60, 61, 64, 72, 74, 79, 80, 81, 83, 94, 95, 97, 98, 100, 101, 102, 103, 105, 106, 109, 118], "diam": [8, 106], "triangul": [8, 9, 74, 107], "shape": [8, 9, 10, 15, 17, 18, 21, 23, 31, 32, 40, 44, 61, 74, 75, 76, 82, 100, 105, 106, 115, 118], "fulfil": [8, 26, 27, 29, 31, 34, 97, 101], "succeq": 8, "sim": [8, 71], "If": [8, 9, 10, 11, 19, 23, 24, 25, 26, 27, 30, 31, 34, 39, 44, 45, 46, 47, 49, 51, 52, 61, 62, 64, 67, 74, 77, 80, 81, 88, 94, 95, 97, 98, 100, 101, 102, 106, 108, 109, 110, 118], "studi": [8, 38, 56, 73, 97], "converg": [8, 25, 26, 29, 30, 31, 36, 37, 39, 46, 48, 49, 54, 63, 73, 75, 77, 80, 96, 97, 98, 100, 107, 112], "famili": 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"Sub-space Correction Methods": [[43, "sub-space-correction-methods"], [44, null]], "Multigrid Methods": [[43, "multigrid-methods"], [44, null]], "Saddle-point Problems": [[43, "saddle-point-problems"], [44, null]], "Non-overlapping Domain Decomposition Methods": [[43, "non-overlapping-domain-decomposition-methods"], [44, null]], "Parallel Solvers": [[43, "parallel-solvers"], [44, null]], "Time-dependent Problems": [[43, "time-dependent-problems"]], "A practical introduction": [[43, "a-practical-introduction"]], "Ordinary differential equations": [[43, "ordinary-differential-equations"]], "Numerical analysis of Parabolic Equations": [[43, "numerical-analysis-of-parabolic-equations"]], "Numerical analysis of Wave Equations": [[43, "numerical-analysis-of-wave-equations"]], "Additional methods": [[43, "additional-methods"]], "Appendix": [[43, "appendix"]], "Some inequalities in Sobolev spaces": [[43, "some-inequalities-in-sobolev-spaces"]], "Literature": [[44, "literature"]], "Installing NGSolve": [[44, "installing-ngsolve"]], "Sobolev Spaces": [[44, null]], "High Order Finite Elements": [[44, null]], "The Chebyshev Method": [[45, "the-chebyshev-method"]], "Chebyshev polynomials": [[45, "chebyshev-polynomials"]], "The Chebyshev iteration": [[45, "the-chebyshev-iteration"]], "The Richardson Iteration": [[46, "the-richardson-iteration"]], "Optimizing the relaxation parameter \\alpha": [[46, "optimizing-the-relaxation-parameter-alpha"]], "Experiments with the Richardson iteration": [[46, "experiments-with-the-richardson-iteration"]], "Conjugate Gradients": [[47, "conjugate-gradients"]], "Solving the minimization problem": [[47, "solving-the-minimization-problem"]], "Expanding the Krylov-space": [[47, "expanding-the-krylov-space"]], "The Gradient Method": [[48, "the-gradient-method"]], "Preconditioning": [[49, "preconditioning"]], "The preconditioned Richardson iteration": [[49, "the-preconditioned-richardson-iteration"]], "The preconditioned gradient method": [[49, "the-preconditioned-gradient-method"]], "Jacobi and Gauss Seidel Preconditioners": [[49, "jacobi-and-gauss-seidel-preconditioners"]], "Basic Iterative Methods": [[50, "basic-iterative-methods"]], "Constrained minimization problem": [[51, "constrained-minimization-problem"]], "Stokes equation within the abstract theory": [[51, "stokes-equation-within-the-abstract-theory"]], "Dirichlet boundary conditions as mixed system": [[51, "dirichlet-boundary-conditions-as-mixed-system"]], "Mixed method for second order equation": [[51, "mixed-method-for-second-order-equation"]], "Abstract theory for mixed finite element methods": [[52, "abstract-theory-for-mixed-finite-element-methods"]], "Finite element error estimates": [[52, "finite-element-error-estimates"]], "Proving discrete LBB by the Fortin operator": [[52, "proving-discrete-lbb-by-the-fortin-operator"]], "Example: Finite elements for Stokes": [[52, "example-finite-elements-for-stokes"]], "Boundary Conditions": [[53, "boundary-conditions"], [71, "boundary-conditions"]], "Parameter Dependent Problems": [[54, "parameter-dependent-problems"], [80, "parameter-dependent-problems"]], "Example: Dirichlet boundary condition by penalty": [[54, "example-dirichlet-boundary-condition-by-penalty"]], "Example: Nearly incompressible materials": [[54, "example-nearly-incompressible-materials"]], "Mixed Methods for second order equations": [[55, "mixed-methods-for-second-order-equations"]], "Stokes Equation": [[56, "stokes-equation"]], "Variational Formulation": [[56, "variational-formulation"]], "Finite Element Spaces": [[56, "finite-element-spaces"]], "\\DeclareMathOperator{\\opdiv}{div}\nHellinger Reissner mixed formulation": [[58, "declaremathoperator-opdiv-div-hellinger-reissner-mixed-formulation"]], "Primal mixed method:": [[58, "primal-mixed-method"]], "Dual mixed method": [[58, "dual-mixed-method"]], "\\DeclareMathOperator{\\opdiv}{div}\n\\DeclareMathOperator{\\opcurl}{curl}\n\\DeclareMathOperator{\\eps}{\\varepsilon}\nReduced symmetry methods": [[59, "declaremathoperator-opdiv-div-declaremathoperator-opcurl-curl-declaremathoperator-eps-varepsilon-reduced-symmetry-methods"]], "Tangential displacement normal normal stress continuous finite elements": [[60, "tangential-displacement-normal-normal-stress-continuous-finite-elements"]], "The divergence of nn-continuous piece-wise smooth functions:": [[60, "the-divergence-of-nn-continuous-piece-wise-smooth-functions"]], "TDNNS Variational formulation:": [[60, "tdnns-variational-formulation"]], "Error estimates:": [[60, "error-estimates"]], "Multigrid and Multilevel Methods": [[61, "multigrid-and-multilevel-methods"]], "Multilevel preconditioner": [[61, "multilevel-preconditioner"]], "Multigrid Preconditioning": [[61, "multigrid-preconditioning"]], "Projection matrices from the finest level": [[61, "projection-matrices-from-the-finest-level"]], "Analysis of the Multigrid Iteration": [[62, "analysis-of-the-multigrid-iteration"]], "The Algorithm": [[62, "the-algorithm"]], "The Approximation Property": [[62, "the-approximation-property"]], "The Smothing Property": [[62, "the-smothing-property"]], "Optimal convergence of the V-cycle": [[62, "optimal-convergence-of-the-v-cycle"]], "Analysis of the multi-level preconditioner": [[63, "analysis-of-the-multi-level-preconditioner"]], "Nearly optimal analysis of the ML - preconditioner": [[63, "nearly-optimal-analysis-of-the-ml-preconditioner"]], "Optimal analysis of the multi-level preconditioner": [[63, "optimal-analysis-of-the-multi-level-preconditioner"]], "Multi-level Extension": [[64, "multi-level-extension"]], "Efficiently computable multi-level decomposition": [[64, "efficiently-computable-multi-level-decomposition"]], "Algorithm": [[64, "algorithm"]], "Extending boundary data": [[64, "extending-boundary-data"]], "The Hellan Herrmann Johnson Method for Kirchhoff plates": [[65, "the-hellan-herrmann-johnson-method-for-kirchhoff-plates"]], "Kirchhoff Plate equation": [[65, "kirchhoff-plate-equation"]], "Hellan-Herrmann-Johnson method": [[65, "hellan-herrmann-johnson-method"]], "Relationship between HHJ and TDNNS": [[66, "relationship-between-hhj-and-tdnns"]], "Commuting diagram for H^1 - H(\\opcurl)": [[66, "commuting-diagram-for-h-1-h-opcurl"]], "Reissner Mindlin Plates": [[67, "reissner-mindlin-plates"]], "Nonlinear Shells": [[68, "nonlinear-shells"]], "Preamble": [[70, "preamble"]], "Natural boundary conditions": [[71, "natural-boundary-conditions"]], "Essential boundary conditions": [[71, "essential-boundary-conditions"]], "3D Solid Mechanics": [[72, "d-solid-mechanics"]], "Linear elasticity": [[72, "linear-elasticity"]], "Variational formulation:": [[72, "variational-formulation"]], "Get started with Netgen-Opencascade geometric modeling": [[73, "get-started-with-netgen-opencascade-geometric-modeling"]], "Experiment with CoefficientFunctions": [[73, "experiment-with-coefficientfunctions"]], "Work with GridFunctions": [[73, "work-with-gridfunctions"]], "Linear Algebra": [[73, "linear-algebra"]], "Experiments with BilienarForms and LinearForms": [[73, "experiments-with-bilienarforms-and-linearforms"]], "Computing dual norms": [[73, "computing-dual-norms"]], "Solving the Poisson Equation": [[74, "solving-the-poisson-equation"]], "Weak formulation": [[74, "weak-formulation"]], "The Sobolev space H^1, linear and bilinear forms": [[74, "the-sobolev-space-h-1-linear-and-bilinear-forms"]], "The Finite Element Method": [[74, "the-finite-element-method"]], "Poisson equation in NGSolve:": [[74, "poisson-equation-in-ngsolve"]], "Visualizing the basis functions": [[74, "visualizing-the-basis-functions"]], "Variable Coefficients": [[76, "variable-coefficients"]], "hp - Finite Elements": [[77, "hp-finite-elements"]], "Legendre Polynomials": [[77, "legendre-polynomials"]], "Error estimate of the L_2 projection": [[77, "error-estimate-of-the-l-2-projection"]], "Orthogonal polynomials on triangles": [[77, "orthogonal-polynomials-on-triangles"]], "Projection based interpolation": [[77, "projection-based-interpolation"]], "The 1D case": [[77, "the-1d-case"]], "Projection based interpolation on triangles": [[77, "projection-based-interpolation-on-triangles"]], "The Bramble-Pasciak Transformation": [[78, "the-bramble-pasciak-transformation"]], "A Small Number of Constraints": [[79, "a-small-number-of-constraints"]], "Projected preconditioner": [[79, "projected-preconditioner"]], "Dirichlet boundary conditions by penalty:": [[80, "dirichlet-boundary-conditions-by-penalty"]], "Penalty formulation for the flux:": [[80, "penalty-formulation-for-the-flux"]], "Maxwell equations:": [[80, "maxwell-equations"]], "Penalty formulation for the Stokes equation:": [[80, "penalty-formulation-for-the-stokes-equation"]], "Robust two-level methods for parameter dependent problems": [[80, "robust-two-level-methods-for-parameter-dependent-problems"]], "Robust smoothers": [[80, "robust-smoothers"]], "Robust coarse-grid correction": [[80, "robust-coarse-grid-correction"]], "Two-level analysis for Maxwell equations": [[80, "two-level-analysis-for-maxwell-equations"]], "Structure of Saddle-point Problems": [[81, "structure-of-saddle-point-problems"]], "Constrained minimization problems": [[81, "constrained-minimization-problems"]], "Examples": [[81, "examples"], [108, "examples"]], "Schur complement iteration": [[81, "schur-complement-iteration"]], "Block-preconditioning": [[81, "block-preconditioning"]], "Finite Element Error Analysis": [[83, "finite-element-error-analysis"]], "Local post-processing": [[83, "local-post-processing"]], "Error Analysis in L_2 \\times H^1": [[84, "error-analysis-in-l-2-times-h-1"]], "Finite Elements in H(\\operatorname{div})": [[85, "finite-elements-in-h-operatorname-div"]], "Piola Transformation": [[85, "piola-transformation"]], "Application of the abstract theory": [[86, "application-of-the-abstract-theory"]], "Dual mixed formulation": [[86, "dual-mixed-formulation"]], "Primal mixed formulation": [[86, "primal-mixed-formulation"]], "The function space H(\\operatorname{div})": [[87, "the-function-space-h-operatorname-div"]], "Normal-trace of functions in H(\\opdiv)": [[87, "normal-trace-of-functions-in-h-opdiv"]], "H(\\opdiv) on sub-domains": [[87, "h-opdiv-on-sub-domains"]], "Hybridization Techniques": [[88, "hybridization-techniques"]], "Friedrichs\u2019 Inequality": [[89, "friedrichs-inequality"], [92, "friedrichs-inequality"]], "The Poincar\u00e9 inequality": [[93, "the-poincare-inequality"]], "The Trace Inequality": [[94, "the-trace-inequality"]], "Numerical verification": [[94, "numerical-verification"]], "Proof of the trace inequality:": [[94, "proof-of-the-trace-inequality"]], "Additive Schwarz Methods": [[102, "additive-schwarz-methods"]], "Block-Jacobi and general additive Schwarz preconditioners": [[102, "block-jacobi-and-general-additive-schwarz-preconditioners"]], "Sub-spaces of {\\mathbb R}^N and of Hilbert-spaces": [[102, "sub-spaces-of-mathbb-r-n-and-of-hilbert-spaces"]], "The Additive Schwarz Lemma": [[102, "the-additive-schwarz-lemma"]], "The upper-bound by the overlap": [[102, "the-upper-bound-by-the-overlap"]], "Some Examples of ASM preconditioners": [[103, "some-examples-of-asm-preconditioners"]], "Diagonal preconditioner for L_2-norm": [[103, "diagonal-preconditioner-for-l-2-norm"]], "Diagonal preconditioner for the H^1 norm": [[103, "diagonal-preconditioner-for-the-h-1-norm"]], "H^1-norm with small L_2-term": [[103, "h-1-norm-with-small-l-2-term"]], "Exercise: Robust preconditioners": [[104, "exercise-robust-preconditioners"]], "Exercise A:": [[104, "exercise-a"]], "Exercise B:": [[104, "exercise-b"]], "Domain Decomposition with minimal overlap": [[105, "domain-decomposition-with-minimal-overlap"]], "Analysis of the method": [[105, "analysis-of-the-method"]], "Adding a coarse grid space": [[105, "adding-a-coarse-grid-space"]], "Graph-based mesh partitioning": [[105, "graph-based-mesh-partitioning"]], "Overlapping Domain Decomposition Methods": [[106, "overlapping-domain-decomposition-methods"]], "Experiments with overlapping DD": [[106, "experiments-with-overlapping-dd"]], "Analysis of the DD preconditioner": [[106, "analysis-of-the-dd-preconditioner"]], "Overlapping DD Methods with coarse grid": [[106, "overlapping-dd-methods-with-coarse-grid"]], "Analysis of the 2-level method:": [[106, "analysis-of-the-2-level-method"]], "Comparison to DD with minimal overlap": [[106, "comparison-to-dd-with-minimal-overlap"]], "Schwarz preconditioners for high order finite elements": [[107, "schwarz-preconditioners-for-high-order-finite-elements"]], "Overlapping blocks": [[107, "overlapping-blocks"]], "Runge Kutta Methods": [[108, "runge-kutta-methods"]], "Butcher tableaus of simple methods": [[108, "butcher-tableaus-of-simple-methods"]], "Convergence of Runge Kutta methods": [[108, "convergence-of-runge-kutta-methods"]], "Explicit methods:": [[108, "explicit-methods"]], "Diagonally implicit Runge-Kutta methods:": [[108, "diagonally-implicit-runge-kutta-methods"]], "Exercise:": [[108, "exercise"], [118, "exercise"]], "Single-step methods": [[109, "single-step-methods"]], "Simple methods based on the integral equation": [[109, "simple-methods-based-on-the-integral-equation"]], "Explicit Euler method (EE)": [[109, "explicit-euler-method-ee"]], "Implicit Euler method (IE)": [[109, "implicit-euler-method-ie"]], "Trapezoidal method": [[109, "trapezoidal-method"]], "Explicit mid-point rule (=improved Euler method = RK2 method)": [[109, "explicit-mid-point-rule-improved-euler-method-rk2-method"]], "Linear stability classification": [[109, "linear-stability-classification"]], "Stability function": [[109, "stability-function"]], "Stability conditions": [[109, "stability-conditions"]], "Single step methods and parabolic equations": [[109, "single-step-methods-and-parabolic-equations"]], "Single step methods and hyperbolic equations": [[109, "single-step-methods-and-hyperbolic-equations"]], "Exercises:": [[109, "exercises"]], "Heat Equation": [[110, "heat-equation"]], "Variational formulation in space": [[110, "variational-formulation-in-space"]], "Galerkin method in space": [[110, "galerkin-method-in-space"]], "Implicit Euler time-stepping": [[110, "implicit-euler-time-stepping"]], "Wave Equation": [[111, "wave-equation"]], "Newmark time-stepping method": [[111, "newmark-time-stepping-method"]], "Exponential Integrators for Parabolic Equations": [[112, "exponential-integrators-for-parabolic-equations"]], "Mass-lumping and Local time-stepping": [[115, "mass-lumping-and-local-time-stepping"]], "The Verlet method:": [[115, "the-verlet-method"]], "Geometry with local details:": [[115, "geometry-with-local-details"]], "Nano-optics: A ring-resonator": [[116, "nano-optics-a-ring-resonator"], [117, "nano-optics-a-ring-resonator"]], "Discontinuous Galerkin for the Wave Equation": [[118, "discontinuous-galerkin-for-the-wave-equation"]], "Testing the differential operators": [[118, "testing-the-differential-operators"]], "Efficient implementation:": [[118, "efficient-implementation"]], "Solving the wave equation:": [[118, "solving-the-wave-equation"]], "Eigenvalues of the discretized Laplace-operator": [[118, "eigenvalues-of-the-discretized-laplace-operator"]], "Equivalent norms on H^1 and on sub-spaces": [[98, "equivalent-norms-on-h-1-and-on-sub-spaces"]], "Generalized derivatives": [[95, "generalized-derivatives"]], "Friedrichs\u2019 inequality": [[99, "friedrichs-inequality"], [100, "friedrichs-inequality"]], "Poincar\u00e9 inequality": [[99, "poincare-inequality"], [100, "poincare-inequality"]], "Friedrichs\u2019 inequality in 1D": [[99, "friedrichs-inequality-in-1d"]], "Friedrichs\u2019 inequality on the square": [[99, "friedrichs-inequality-on-the-square"]], "Poincar\u00e9 inequality in 1D": [[99, "poincare-inequality-in-1d"]], "Bramble-Hilbert Lemma": [[99, "bramble-hilbert-lemma"]], "Mean-value interpolation": [[99, "mean-value-interpolation"]], "Scaled domain": [[99, "scaled-domain"]], "Fractional Sobolev spaces": [[99, "fractional-sobolev-spaces"]], "Step function": [[99, "step-function"]], "Point evaluation functional": [[99, "point-evaluation-functional"]], "Trace theorems and their applications": [[97, "trace-theorems-and-their-applications"]], "Integration by parts": [[97, "integration-by-parts"]], "Sobolev spaces over sub-domains": [[97, "sobolev-spaces-over-sub-domains"]], "Extension operators": [[97, "extension-operators"]], "The trace space H^{1/2}": [[97, "the-trace-space-h-1-2"]], "Interpolation spaces": [[97, "interpolation-spaces"]], "General definition:": [[97, "general-definition"]], "The trace space on one edge of the square": [[97, "the-trace-space-on-one-edge-of-the-square"]], "Sobolev spaces": [[96, "sobolev-spaces"]], "The weak formulation of the Poisson equation": [[101, "the-weak-formulation-of-the-poisson-equation"]], "Shift theorems": [[101, "shift-theorems"]], "Non-linear dynamics": [[57, "non-linear-dynamics"]], "3D-TDNNS Elasticity and Reissner Mindlin Plate elements": [[69, "d-tdnns-elasticity-and-reissner-mindlin-plate-elements"]], "Experiments with norms": [[100, "experiments-with-norms"]], "Inverse estimates": [[100, "inverse-estimates"]], "Trace inequality": [[100, "trace-inequality"]], "Korn\u2019s inequality": [[100, "korn-s-inequality"]]}, "indexentries": {}}) \ No newline at end of file +Search.setIndex({"docnames": ["DG/Nitsche", "DG/elliptic", "DG/elliptic_stdDG", "DG/fourthorder", "DG/instationary", "DG/splitting", "DG/stationary", "DG/stokes", "FEM/erroranalysis", "FEM/finiteelements", "FEM/myFiniteElement", "FEM/nonconforming", "FEM/sytemassembling", "MPIparallel/PETSc_interface", "MPIparallel/distmesh", "MPIparallel/hdivnstsimple/NavierStokesMT", "MPIparallel/hdivnstsimple/TestConvert", 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"subspacecorrection/examples.ipynb", "subspacecorrection/exercise.ipynb", "subspacecorrection/minimaldd.ipynb", "subspacecorrection/overlapping.ipynb", "subspacecorrection/pversion.ipynb", "timedependent/ODEs/runge_kutta.ipynb", "timedependent/ODEs/singlestep.ipynb", "timedependent/intro/heatequation.ipynb", "timedependent/intro/waveequation.ipynb", "timedependent/parabolic/exponential.ipynb", "timedependent/waves/Maxwell-DG.ipynb", "timedependent/waves/Untitled.ipynb", "timedependent/waves/lts.ipynb", "timedependent/waves/ringresonator.ipynb", "timedependent/waves/testpml/ringresonator.ipynb", "timedependent/waves/wave-leapfrogDG.ipynb"], "titles": ["38. Nitsche\u2019s Method for boundary and interface conditions", "40. Hybrid DG for elliptic equations", "DG - Methods for elliptic problems", "42. Fourth Order Equation", "37. Instationary Transport Equation", "41. Splitting Methods for the time-dependent convection diffusion equation", "36. Stationary Transport Equation", "43. H(div)-conforming Stokes", "22. Finite element error analysis", "20. Finite Element Method", "24. Implementation of Finite Elements", "23. Non-conforming Finite Element Methods", "21. Finite element system assembling", "84. NGSolve - PETSc interface", "79. Distributed Meshes and Spaces", "<no title>", "<no title>", "<no title>", "<no title>", "78. Introduction to MPI with mpi4py", "82. Iteration methods in parallel", "85. Solving Stokes in parallel", "83. Using PETSc", "81. Consistent and Distributed Vectors", "This repository contains an interactive introduction to the Finite Element Method", "7. Basic properties", "10. Coercive variational problems and their approximation", "9. Riesz representation theorem and symmetric variational problems", "12. Exercises", "11. Inf-sup stable variational problems", "8. Projection onto subspaces", "25. A posteriori error estimates", "28. Equilibrated Residual Error Estimates", "27. Goal driven error estimates", "26. The residual error estimator", "77. BDDC - Preconditioner", "75. FETI methods", "76. FETI-DP", "72. Introduction to Non-overlapping Domain Decomposition", "73. Traces spaces", "Implement a parallel BDDC preconditioner", "Helmholtz Equation", "Grating", "An Interactive Introduction to the Finite Element Method", "An Interactive Introduction to the Finite Element Method", "54. The Chebyshev Method", "51. The Richardson Iteration", "55. Conjugate Gradients", "52. The Gradient Method", "53. Preconditioning", "50. Basic Iterative Methods", "33. Abstract Theory", "34. Abstract theory for mixed finite element methods", "31. Boundary Conditions", "35. Parameter Dependent Problems", "32. Mixed Methods for second order equations", "30. Stokes Equation", "Non-linear dynamics", "\\(\\DeclareMathOperator{\\opdiv}{div}\\)\nHellinger Reissner mixed formulation", "\\(\\DeclareMathOperator{\\opdiv}{div}\\)\n\\(\\DeclareMathOperator{\\opcurl}{curl}\\)\n\\(\\DeclareMathOperator{\\eps}{\\varepsilon}\\)\nReduced symmetry methods", "Tangential displacement normal normal stress continuous finite elements", "62. Multigrid and Multilevel Methods", "64. Analysis of the Multigrid Iteration", "63. Analysis of the multi-level preconditioner", "65. Multi-level Extension", "The Hellan Herrmann Johnson Method for Kirchhoff plates", "Relationship between HHJ and TDNNS", "Reissner Mindlin Plates", "Nonlinear Shells", "3D-TDNNS Elasticity and Reissner Mindlin Plate elements", "Preamble", "2. Boundary Conditions", "5. 3D Solid Mechanics", "6. Exercises", "1. Solving the Poisson Equation", "4. Iterative Solvers", "3. Variable Coefficients", "29. hp - Finite Elements", "67. The Bramble-Pasciak Transformation", "68. A Small Number of Constraints", "70. Parameter Dependent Problems", "66. Structure of Saddle-point Problems", "<no title>", "47. Finite Element Error Analysis", "48. Error Analysis in \\(L_2 \\times H^1\\)", "46. Finite Elements in \\(H(\\operatorname{div})\\)", "44. Application of the abstract theory", "45. The function space \\(H(\\operatorname{div})\\)", "49. Hybridization Techniques", "Friedrichs\u2019 Inequality", "<no title>", "<no title>", "Friedrichs\u2019 Inequality", "The Poincar\u00e9 inequality", "The Trace Inequality", "13. Generalized derivatives", "14. Sobolev spaces", "15. Trace theorems and their applications", "16. Equivalent norms on \\(H^1\\) and on sub-spaces", "19. Exercises", "18. Experiments with norms", "17. The weak formulation of the Poisson equation", "56. Additive Schwarz Methods", "57. Some Examples of ASM preconditioners", "61. Exercise: Robust preconditioners", "59. Domain Decomposition with minimal overlap", "60. Overlapping Domain Decomposition Methods", "58. Schwarz preconditioners for high order finite elements", "Runge Kutta Methods", "Single-step methods", "Heat Equation", "Wave Equation", "Exponential Integrators for Parabolic Equations", "<no title>", "<no title>", "Mass-lumping and Local time-stepping", "Nano-optics: A ring-resonator", "Nano-optics: A ring-resonator", "Discontinuous Galerkin for the Wave Equation"], "terms": {"from": [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 25, 26, 27, 29, 31, 32, 34, 35, 36, 37, 39, 40, 41, 42, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 60, 62, 63, 64, 65, 67, 68, 69, 70, 71, 72, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 85, 86, 87, 88, 89, 93, 94, 95, 97, 98, 100, 102, 103, 105, 106, 107, 108, 109, 110, 111, 112, 113, 114, 115, 116, 117, 118], "netgen": [0, 1, 2, 3, 4, 5, 6, 7, 14, 15, 17, 18, 21, 35, 36, 37, 40, 41, 42, 44, 45, 53, 55, 56, 60, 61, 62, 64, 65, 67, 68, 70, 72, 74, 75, 76, 78, 79, 80, 81, 82, 83, 85, 86, 88, 100, 105, 110, 111, 112, 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102], "uri": 5, "ascher": 5, "steven": 5, "j": [5, 9, 12, 19, 28, 29, 32, 35, 36, 37, 40, 44, 46, 47, 49, 62, 65, 74, 77, 80, 85, 96, 97, 102, 105, 106, 108, 109, 110, 111, 112, 115, 116, 117, 118], "ruuth": 5, "raymond": 5, "spiteri": 5, "numer": [5, 11, 12, 34, 44, 47, 50, 74, 80, 81, 83, 100, 103, 108, 109, 112, 118], "mathemat": [5, 32, 44, 74, 100], "25": [5, 14, 20, 36, 37, 46, 72, 75, 78, 80, 81, 82, 85, 88, 105, 107], "1997": 5, "text": [6, 28, 32, 41, 45, 46, 47, 48, 49, 51, 53, 55, 56, 58, 59, 60, 61, 63, 65, 66, 68, 71, 72, 74, 77, 79, 80, 81, 85, 86, 87, 97, 99, 102, 103, 104, 105, 106, 108, 109, 110, 111, 112], "model": [6, 56, 67, 68, 71, 74, 75, 76, 115], "inject": [6, 28, 29, 97], "concentr": 6, "flow": [6, 26, 81, 109], "exampl": [6, 9, 11, 19, 25, 26, 29, 32, 39, 43, 44, 50, 51, 61, 71, 74, 79, 80, 82, 95, 100, 105, 107, 110, 111], "milk": 6, "coffe": 6, "u_": [6, 23, 29, 32, 34, 40, 51, 62, 63, 74, 89, 92, 94, 101, 103, 112], "inflow": [6, 15, 17, 18], "popular": 6, "Their": 6, "strength": 6, "techniqu": [6, 33, 38, 43, 44, 62, 77, 97, 98, 102, 109], "stabl": [6, 28, 43, 44, 52, 58, 64, 77, 80, 106, 108, 109], "combin": [6, 8, 10, 11, 14, 28, 34, 47, 49, 56, 61, 62, 63, 74, 97, 102, 107, 112, 118], "seen": [6, 47, 51, 61, 62, 80, 88, 96], "extens": [6, 11, 22, 25, 35, 43, 44, 61, 77, 88], "trial": [6, 10, 74, 76], "piecewis": [6, 31, 34, 97, 105, 106], "follow": [6, 8, 10, 11, 29, 31, 32, 33, 34, 35, 38, 39, 44, 45, 49, 51, 52, 61, 62, 63, 64, 71, 77, 80, 84, 85, 88, 94, 95, 96, 97, 98, 100, 102, 103, 104, 105, 106, 108, 118], "wise": [6, 14, 23, 32, 35, 56, 58, 68, 74, 83, 85, 103, 118], "smooth": [6, 8, 9, 12, 31, 39, 58, 61, 62, 87, 95, 96, 97, 101, 103], "evei": 6, "sum_": [6, 8, 9, 11, 12, 23, 31, 32, 33, 34, 36, 38, 39, 40, 45, 47, 49, 60, 63, 64, 74, 77, 84, 87, 88, 96, 97, 98, 99, 101, 102, 103, 108, 110, 118], "subset": [6, 8, 9, 11, 12, 14, 25, 27, 29, 32, 34, 35, 38, 39, 45, 47, 52, 61, 62, 64, 65, 66, 74, 77, 80, 81, 85, 89, 92, 94, 95, 97, 98, 99, 101, 102, 103, 106, 108, 109, 110], "b_n": 6, "outward": 6, "decis": 6, "blow": 6, "up": [6, 8, 11, 13, 14, 23, 26, 28, 29, 47, 49, 56, 61, 64, 77, 80, 94, 95, 96, 98, 106, 108, 109, 110, 118], "cl": [6, 11, 34, 95, 99], "u_t": [6, 60], "outflow": [6, 15, 17, 18, 55], "v_h": [6, 8, 11, 12, 23, 26, 27, 29, 32, 33, 34, 52, 60, 63, 66, 74, 80, 81, 83, 84, 88, 100, 102, 103, 106, 109, 110], "u_h": [6, 8, 11, 12, 26, 27, 29, 31, 32, 33, 34, 52, 60, 63, 74, 80, 83, 84, 88, 100, 102, 106, 109, 110], "all": [6, 8, 9, 10, 14, 19, 23, 25, 27, 28, 29, 30, 31, 32, 34, 35, 37, 44, 47, 49, 50, 51, 52, 61, 63, 64, 67, 70, 71, 72, 74, 80, 87, 96, 97, 98, 99, 101, 102, 103, 105, 106, 107, 108, 109, 110, 112, 118], "6": [6, 13, 20, 22, 36, 37, 38, 42, 45, 46, 53, 61, 68, 72, 74, 75, 76, 77, 78, 79, 80, 81, 82, 85, 88, 100, 105, 107, 108, 109, 112, 115], "28": [6, 20, 36, 37, 46, 74, 75, 78, 80, 81, 82, 107], "hybrid": [7, 43, 44, 52, 65, 70], "lehrenfeld": [7, 70], "sch\u00f6berl": [7, 43, 44, 60, 69, 70, 80, 116, 117], "2016": 7, "bdm": [7, 32, 85], "k": [7, 8, 25, 26, 32, 39, 45, 46, 47, 48, 49, 56, 63, 64, 65, 77, 81, 83, 84, 85, 88, 95, 96, 97, 98, 99, 101, 109], "k_": 7, "splinegeometri": [7, 42, 60, 78, 82, 86], "geo": [7, 35, 36, 37, 42, 60, 68, 72, 76, 78, 79, 80, 82, 86, 105, 115, 118], "addrectangl": [7, 42, 60, 78, 82, 86], "41": [7, 14, 15, 17, 18, 20, 21, 36, 37, 46, 74, 75, 78, 80, 81, 82, 107], "wall": [7, 15, 17, 18, 21, 78, 82], "outlet": [7, 15, 17, 18, 21, 78, 82], "inlet": [7, 15, 17, 18, 21, 78, 82], "addcircl": [7, 42, 78, 82, 86], "leftdomain": [7, 42, 78, 82, 86], "rightdomain": [7, 42, 78, 82, 86], "cyl": [7, 15, 21, 68, 72, 75, 78, 82], "08": [7, 20, 36, 37, 46, 61, 75, 78, 80, 81, 82, 88, 107, 112], "0x105d6ea50": [], "vt": 7, "vf": 7, "tangentialfacetfespac": [7, 16, 17, 18, 113], "q": [7, 9, 21, 25, 32, 51, 52, 53, 54, 56, 60, 76, 77, 78, 81, 82, 97, 98, 118], "nu": [7, 15, 17, 18, 72, 80], "tang": [7, 16, 17, 18, 60, 65, 67, 68], "thesi": [7, 57, 60, 70], "christoph": [7, 70], "page": [7, 45, 47], "71": [7, 36, 37, 46, 75, 80, 82, 107], "invstok": 7, "uin": [7, 15, 17, 18, 21, 78, 82], "re": [7, 32, 34, 46, 48, 49, 61, 79, 110], "vel": 7, "veloc": [7, 15, 56, 57, 81, 111], "pressur": [7, 56, 80, 81], "let": [8, 11, 12, 19, 24, 25, 26, 27, 28, 29, 30, 31, 34, 35, 38, 39, 45, 46, 47, 48, 60, 61, 62, 63, 64, 74, 77, 79, 80, 81, 83, 89, 92, 94, 95, 96, 97, 98, 99, 101, 102, 103, 105, 106, 107, 110], "variat": [8, 28, 43, 44, 51, 58, 65, 71, 73, 74, 76, 79, 80, 81, 83, 84, 88, 98, 101, 102, 111], "problem": [8, 10, 11, 12, 20, 32, 33, 35, 36, 37, 38, 39, 45, 48, 49, 58, 61, 62, 63, 65, 67, 68, 73, 74, 75, 77, 79, 83, 86, 88, 94, 97, 98, 100, 101, 102, 103, 104, 105, 106, 109], "sub": [8, 11, 22, 23, 25, 28, 30, 32, 35, 36, 37, 38, 40, 47, 51, 52, 61, 63, 71, 74, 75, 77, 79, 80, 81, 85, 95, 100, 101, 103, 105, 106, 108, 109, 110], "cea": [8, 11, 62], "": [8, 9, 11, 17, 18, 24, 25, 26, 27, 28, 30, 32, 37, 43, 44, 51, 52, 54, 62, 68, 69, 70, 72, 73, 74, 75, 79, 81, 89, 97, 98, 101, 102, 108, 109, 112, 115, 118], "lemma": [8, 9, 25, 26, 29, 30, 34, 35, 43, 61, 62, 63, 77, 80, 97, 98, 103, 105, 106, 109], "bound": [8, 9, 11, 25, 26, 28, 29, 31, 32, 33, 34, 45, 46, 47, 49, 63, 80, 81, 94, 95, 96, 97, 98, 99, 100, 101, 103, 105, 106], "best": [8, 26, 52, 65, 77, 107], "_v": [8, 11, 25, 26, 27, 28, 29, 31, 33, 34, 51, 52, 73, 97, 98, 101], "leq": [8, 9, 11, 25, 26, 27, 28, 29, 30, 31, 32, 34, 35, 45, 46, 47, 48, 49, 51, 52, 59, 62, 63, 64, 65, 69, 77, 80, 81, 83, 87, 89, 92, 93, 94, 95, 96, 97, 98, 99, 100, 101, 102, 103, 105, 106, 109, 112], "inf_": [8, 11, 26, 28, 29, 30, 35, 39, 46, 52, 60, 63, 65, 77, 80, 97, 98, 102, 103, 105, 106], "constant": [8, 9, 10, 11, 14, 26, 27, 28, 31, 32, 34, 38, 40, 49, 52, 56, 57, 61, 62, 64, 65, 74, 76, 77, 85, 89, 95, 97, 98, 99, 100, 103, 105, 106, 108, 109, 118], "factor": [8, 26, 29, 46, 61, 62, 64, 68, 74, 77, 80, 81, 94, 95, 97, 98, 105, 106, 118], "ratio": 8, "take": [8, 13, 24, 27, 29, 30, 31, 34, 38, 39, 44, 52, 60, 63, 74, 76, 81, 84, 86, 96, 97, 101, 103, 105, 109], "interpol": [8, 9, 11, 16, 34, 44, 52, 63, 64, 71, 72, 73, 74, 80, 83, 106, 115, 118], "i_": [8, 9, 102], "cal": [8, 9, 11, 12, 31, 33, 34, 95, 96], "dimension": [8, 10, 25, 26, 27, 28, 29, 47, 50, 74, 77, 81, 84, 97, 98, 100, 103, 112], "domain": [8, 9, 10, 11, 23, 31, 35, 36, 37, 40, 41, 51, 61, 63, 64, 71, 74, 79, 80, 94, 95, 96, 98, 100, 101, 110, 116, 117], "relat": [8, 12, 27, 45, 48, 62, 76, 98], "affin": [8, 9, 58], "transform": [8, 9, 10, 11, 43, 44, 49, 74, 79, 82, 94, 97, 103], "f_t": [8, 9, 11, 12, 23, 32, 60], "rightarrow": [8, 9, 11, 25, 26, 27, 28, 29, 30, 32, 33, 34, 35, 39, 48, 51, 52, 54, 56, 64, 66, 67, 72, 73, 74, 76, 77, 80, 94, 95, 96, 97, 98, 99, 100, 101, 102, 103, 106, 109, 110, 112], "mathbb": [8, 9, 11, 12, 25, 26, 28, 29, 30, 33, 34, 35, 46, 47, 48, 49, 50, 61, 64, 72, 73, 74, 76, 79, 81, 89, 92, 94, 95, 96, 97, 98, 99, 100, 101, 103, 108, 109, 112, 118], "regular": [8, 9, 10, 31, 32, 37, 51, 60, 62, 63, 74, 80, 81, 88, 95, 101, 102, 106], "Then": [8, 11, 12, 14, 19, 23, 25, 26, 27, 28, 29, 30, 32, 34, 36, 37, 39, 44, 45, 46, 48, 51, 52, 61, 62, 63, 64, 77, 80, 81, 89, 94, 95, 96, 97, 98, 101, 102, 103, 105, 109, 115], "hold": [8, 9, 11, 25, 26, 27, 29, 30, 31, 32, 39, 47, 48, 49, 51, 61, 62, 63, 64, 72, 74, 77, 80, 87, 94, 95, 96, 97, 98, 100, 101, 102, 103, 106, 109], "det": [8, 85, 97, 103, 109, 118], "align": [8, 9, 12, 26, 27, 29, 45, 46, 48, 49, 50, 77, 95, 97, 98, 101], "x_": [8, 11, 47], "i_m": 8, "ldot": [8, 9, 12, 28, 32, 45, 46, 47, 48, 49, 61, 64, 74, 77, 87, 95, 97, 101, 102, 108, 109, 110, 112], "i_1": [8, 39], "j_m": 8, "j_1": 8, "b_": [8, 51, 108], "equat": [8, 11, 12, 26, 29, 31, 32, 33, 34, 36, 38, 39, 46, 50, 52, 53, 59, 63, 71, 72, 75, 76, 77, 79, 81, 86, 88, 97, 98, 102, 108, 115, 116, 117], "preceq": [8, 11, 33, 34, 62, 63, 64, 77, 80, 97, 98, 101, 103, 105], "proof": [8, 11, 25, 26, 27, 29, 30, 32, 34, 35, 52, 62, 63, 64, 73, 77, 80, 81, 84, 85, 87, 89, 95, 96, 97, 98, 101, 102, 103, 105, 106, 109], "chain": [8, 85, 97, 103], "box": [8, 11, 21, 25, 26, 27, 29, 30, 63, 72, 75, 76, 77, 80, 82, 94, 96, 97, 98, 101, 102, 105, 118], "diamet": 8, "h_t": [8, 11, 32, 34], "operatornam": [8, 23, 25, 26, 27, 29, 36, 39, 44, 47, 51, 52, 54, 55, 56, 57, 59, 60, 61, 64, 72, 74, 79, 80, 81, 83, 94, 95, 97, 98, 100, 101, 102, 103, 105, 106, 109, 118], "diam": [8, 106], "triangul": [8, 9, 74, 107], "shape": [8, 9, 10, 15, 17, 18, 21, 23, 31, 32, 40, 44, 61, 74, 75, 76, 82, 100, 105, 106, 115, 118], "fulfil": [8, 26, 27, 29, 31, 34, 97, 101], "succeq": 8, "sim": [8, 71], "If": [8, 9, 10, 11, 19, 23, 24, 25, 26, 27, 30, 31, 34, 39, 44, 45, 46, 47, 49, 51, 52, 61, 62, 64, 67, 74, 77, 80, 81, 88, 94, 95, 97, 98, 100, 101, 102, 106, 108, 109, 110, 118], "studi": [8, 38, 56, 73, 97], "converg": [8, 25, 26, 29, 30, 31, 36, 37, 39, 46, 48, 49, 54, 63, 73, 75, 77, 80, 96, 97, 98, 100, 107, 112], "famili": [8, 85, 97], "decreas": [8, 38, 46, 49], "case": [8, 9, 12, 22, 26, 27, 28, 29, 33, 34, 49, 52, 56, 58, 63, 65, 67, 71, 74, 76, 80, 81, 96, 97, 98, 101, 102, 103, 112], "geq": [8, 11, 25, 26, 28, 29, 30, 31, 32, 45, 49, 51, 52, 62, 64, 65, 77, 81, 84, 99, 101], "refer": [8, 9, 10, 11, 34, 50, 60, 62, 63, 85, 103, 109, 118], "triangl": [8, 9, 31, 32, 34, 52, 64, 80, 83, 103, 115], "b_t": 8, "simeq": [8, 11, 34, 97, 98, 100, 101], "main": [8, 32], "applic": [8, 11, 31, 33, 43, 44, 47, 61, 70, 78, 79, 118], "brambl": [8, 11, 34, 43, 44, 62, 77, 82, 98], "hilbert": [8, 11, 25, 26, 27, 28, 29, 30, 34, 51, 52, 54, 74, 77, 80, 96, 97, 98, 101, 110], "sometim": 8, "itself": [8, 14, 19, 102, 111], "v_t": [8, 9, 23, 32, 60, 85], "psi_t": [8, 9], "contain": [8, 9, 22, 25, 31, 32, 34, 39, 47, 74, 80, 95, 96, 101, 106, 109], "i_t": [8, 9], "first": [8, 10, 12, 24, 25, 27, 30, 31, 34, 36, 44, 45, 46, 49, 51, 52, 53, 54, 55, 56, 59, 60, 62, 63, 68, 74, 77, 79, 80, 81, 86, 88, 94, 95, 97, 98, 100, 101, 103, 106, 109, 111, 112, 115, 118], "prove": [8, 11, 27, 28, 30, 32, 46, 49, 51, 62, 63, 77, 80, 81, 94, 96, 97, 98, 99, 100, 101, 102, 103, 106], "psi_": [8, 9, 62], "varphi_": [8, 62, 64, 77, 87, 103], "project": [8, 9, 26, 27, 35, 43, 44, 62, 63, 80, 83, 105, 106, 110, 115], "l": [8, 11, 13, 14, 17, 18, 22, 25, 26, 27, 29, 44, 45, 49, 51, 61, 62, 63, 64, 66, 75, 77, 80, 81, 83, 86, 96, 97, 98, 100, 101, 102, 106, 108, 109, 115], "result": [8, 11, 12, 14, 19, 23, 35, 49, 60, 63, 64, 67, 69, 74, 77, 80, 94, 96, 97, 98, 103, 106, 107], "theorem": [8, 11, 26, 28, 29, 30, 34, 35, 43, 44, 51, 52, 62, 63, 64, 74, 77, 80, 81, 87, 96, 98, 99, 100, 102, 106], "v_": [8, 9, 11, 12, 29, 32, 40, 51, 61, 62, 65, 77, 80, 97, 100], "local": [8, 9, 10, 12, 14, 20, 22, 23, 31, 32, 33, 34, 35, 38, 40, 43, 44, 57, 61, 62, 64, 74, 75, 77, 78, 80, 95, 96, 97, 106], "each": [8, 10, 11, 12, 14, 23, 31, 34, 49, 56, 61, 74, 79, 80, 88, 95, 96, 97, 105], "back": [8, 14, 19, 35, 45, 49, 56, 59, 81, 97, 99, 106, 109, 118], "individu": [8, 10, 38, 72, 74, 97, 102, 105, 108], "quasi": [8, 26, 29, 34, 52, 64, 80, 106], "uniform": [8, 11, 14, 31, 46, 63, 75], "essenti": [8, 32, 53, 63, 86, 95, 97, 100, 105], "eqnarrai": [8, 11, 23, 31, 32, 34, 35, 39, 45, 48, 49, 51, 52, 56, 58, 60, 61, 62, 63, 64, 65, 76, 77, 80, 81, 83, 84, 86, 87, 89, 94, 96, 97, 102, 103, 106, 108, 109, 110, 111, 112, 118], "interest": [8, 49, 54, 67, 97], "rate": [8, 26, 46, 75, 77], "assum": [8, 10, 11, 12, 23, 26, 27, 28, 29, 32, 47, 48, 50, 51, 53, 62, 63, 71, 74, 80, 87, 95, 96, 97, 98, 100, 101, 102, 103, 105, 106, 108, 109], "weak": [8, 27, 29, 41, 44, 54, 55, 56, 71, 76, 80, 85, 87, 95, 96, 97, 103, 110], "abov": [8, 9, 11, 26, 28, 29, 33, 34, 49, 63, 73, 74, 83, 100, 102, 105, 106, 109], "obtain": [8, 11, 12, 14, 25, 26, 29, 30, 31, 33, 36, 39, 45, 46, 47, 48, 49, 56, 59, 61, 63, 64, 65, 74, 77, 78, 80, 81, 85, 86, 88, 94, 97, 98, 100, 103, 105, 106, 109, 110, 111, 112, 118], "natur": [8, 23, 34, 44, 53, 56, 60, 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"boundary-conditions"], [71, "boundary-conditions"]], "Parameter Dependent Problems": [[54, "parameter-dependent-problems"], [80, "parameter-dependent-problems"]], "Example: Dirichlet boundary condition by penalty": [[54, "example-dirichlet-boundary-condition-by-penalty"]], "Example: Nearly incompressible materials": [[54, "example-nearly-incompressible-materials"]], "Mixed Methods for second order equations": [[55, "mixed-methods-for-second-order-equations"]], "Stokes Equation": [[56, "stokes-equation"]], "Variational Formulation": [[56, "variational-formulation"]], "Finite Element Spaces": [[56, "finite-element-spaces"]], "\\DeclareMathOperator{\\opdiv}{div}\nHellinger Reissner mixed formulation": [[58, "declaremathoperator-opdiv-div-hellinger-reissner-mixed-formulation"]], "Primal mixed method:": [[58, "primal-mixed-method"]], "Dual mixed method": [[58, "dual-mixed-method"]], 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"projection-matrices-from-the-finest-level"]], "Analysis of the Multigrid Iteration": [[62, "analysis-of-the-multigrid-iteration"]], "The Algorithm": [[62, "the-algorithm"]], "The Approximation Property": [[62, "the-approximation-property"]], "The Smothing Property": [[62, "the-smothing-property"]], "Optimal convergence of the V-cycle": [[62, "optimal-convergence-of-the-v-cycle"]], "Analysis of the multi-level preconditioner": [[63, "analysis-of-the-multi-level-preconditioner"]], "Nearly optimal analysis of the ML - preconditioner": [[63, "nearly-optimal-analysis-of-the-ml-preconditioner"]], "Optimal analysis of the multi-level preconditioner": [[63, "optimal-analysis-of-the-multi-level-preconditioner"]], "Multi-level Extension": [[64, "multi-level-extension"]], "Efficiently computable multi-level decomposition": [[64, "efficiently-computable-multi-level-decomposition"]], "Algorithm": [[64, "algorithm"]], "Extending boundary data": [[64, "extending-boundary-data"]], "The Hellan Herrmann Johnson Method for Kirchhoff plates": [[65, "the-hellan-herrmann-johnson-method-for-kirchhoff-plates"]], "Kirchhoff Plate equation": [[65, "kirchhoff-plate-equation"]], "Hellan-Herrmann-Johnson method": [[65, "hellan-herrmann-johnson-method"]], "Relationship between HHJ and TDNNS": [[66, "relationship-between-hhj-and-tdnns"]], "Commuting diagram for H^1 - H(\\opcurl)": [[66, "commuting-diagram-for-h-1-h-opcurl"]], "Reissner Mindlin Plates": [[67, "reissner-mindlin-plates"]], "Nonlinear Shells": [[68, "nonlinear-shells"]], "Preamble": [[70, "preamble"]], "Natural boundary conditions": [[71, "natural-boundary-conditions"]], "Essential boundary conditions": [[71, "essential-boundary-conditions"]], "3D Solid Mechanics": [[72, "d-solid-mechanics"]], "Linear elasticity": [[72, "linear-elasticity"]], "Variational formulation:": [[72, "variational-formulation"]], "Get started with Netgen-Opencascade geometric modeling": [[73, "get-started-with-netgen-opencascade-geometric-modeling"]], "Experiment with CoefficientFunctions": [[73, "experiment-with-coefficientfunctions"]], "Work with GridFunctions": [[73, "work-with-gridfunctions"]], "Linear Algebra": [[73, "linear-algebra"]], "Experiments with BilienarForms and LinearForms": [[73, "experiments-with-bilienarforms-and-linearforms"]], "Computing dual norms": [[73, "computing-dual-norms"]], "Solving the Poisson Equation": [[74, "solving-the-poisson-equation"]], "Weak formulation": [[74, "weak-formulation"]], "The Sobolev space H^1, linear and bilinear forms": [[74, "the-sobolev-space-h-1-linear-and-bilinear-forms"]], "The Finite Element Method": [[74, "the-finite-element-method"]], "Poisson equation in NGSolve:": [[74, "poisson-equation-in-ngsolve"]], "Visualizing the basis functions": [[74, "visualizing-the-basis-functions"]], "Variable Coefficients": [[76, "variable-coefficients"]], "hp - Finite Elements": [[77, "hp-finite-elements"]], "Legendre Polynomials": [[77, "legendre-polynomials"]], "Error estimate of the L_2 projection": [[77, "error-estimate-of-the-l-2-projection"]], "Orthogonal polynomials on triangles": [[77, "orthogonal-polynomials-on-triangles"]], "Projection based interpolation": [[77, "projection-based-interpolation"]], "The 1D case": [[77, "the-1d-case"]], "Projection based interpolation on triangles": [[77, "projection-based-interpolation-on-triangles"]], "The Bramble-Pasciak Transformation": [[78, "the-bramble-pasciak-transformation"]], "A Small Number of Constraints": [[79, "a-small-number-of-constraints"]], "Projected preconditioner": [[79, "projected-preconditioner"]], "Dirichlet boundary conditions by penalty:": [[80, "dirichlet-boundary-conditions-by-penalty"]], "Penalty formulation for the flux:": [[80, "penalty-formulation-for-the-flux"]], "Maxwell equations:": [[80, "maxwell-equations"]], "Penalty formulation for the Stokes equation:": [[80, "penalty-formulation-for-the-stokes-equation"]], "Robust two-level methods for parameter dependent problems": [[80, "robust-two-level-methods-for-parameter-dependent-problems"]], "Robust smoothers": [[80, "robust-smoothers"]], "Robust coarse-grid correction": [[80, "robust-coarse-grid-correction"]], "Two-level analysis for Maxwell equations": [[80, "two-level-analysis-for-maxwell-equations"]], "Structure of Saddle-point Problems": [[81, "structure-of-saddle-point-problems"]], "Constrained minimization problems": [[81, "constrained-minimization-problems"]], "Examples": [[81, "examples"], [108, "examples"]], "Schur complement iteration": [[81, "schur-complement-iteration"]], "Block-preconditioning": [[81, "block-preconditioning"]], "Finite Element Error Analysis": [[83, "finite-element-error-analysis"]], "Local post-processing": [[83, "local-post-processing"]], "Error Analysis in L_2 \\times H^1": [[84, "error-analysis-in-l-2-times-h-1"]], "Finite Elements in H(\\operatorname{div})": [[85, "finite-elements-in-h-operatorname-div"]], "Piola Transformation": [[85, "piola-transformation"]], "Application of the abstract theory": [[86, "application-of-the-abstract-theory"]], "Dual mixed formulation": [[86, "dual-mixed-formulation"]], "Primal mixed formulation": [[86, "primal-mixed-formulation"]], "The function space H(\\operatorname{div})": [[87, "the-function-space-h-operatorname-div"]], "Normal-trace of functions in H(\\opdiv)": [[87, "normal-trace-of-functions-in-h-opdiv"]], "H(\\opdiv) on sub-domains": [[87, "h-opdiv-on-sub-domains"]], "Hybridization Techniques": [[88, "hybridization-techniques"]], "Friedrichs\u2019 Inequality": [[89, "friedrichs-inequality"], [92, "friedrichs-inequality"]], "The Poincar\u00e9 inequality": [[93, "the-poincare-inequality"]], "The Trace Inequality": [[94, "the-trace-inequality"]], "Numerical verification": [[94, "numerical-verification"]], "Proof of the trace inequality:": [[94, "proof-of-the-trace-inequality"]], "Additive Schwarz Methods": [[102, "additive-schwarz-methods"]], "Block-Jacobi and general additive Schwarz preconditioners": [[102, "block-jacobi-and-general-additive-schwarz-preconditioners"]], "Sub-spaces of {\\mathbb R}^N and of Hilbert-spaces": [[102, "sub-spaces-of-mathbb-r-n-and-of-hilbert-spaces"]], "The Additive Schwarz Lemma": [[102, "the-additive-schwarz-lemma"]], "The upper-bound by the overlap": [[102, "the-upper-bound-by-the-overlap"]], "Some Examples of ASM preconditioners": [[103, "some-examples-of-asm-preconditioners"]], "Diagonal preconditioner for L_2-norm": [[103, "diagonal-preconditioner-for-l-2-norm"]], "Diagonal preconditioner for the H^1 norm": [[103, "diagonal-preconditioner-for-the-h-1-norm"]], "H^1-norm with small L_2-term": [[103, "h-1-norm-with-small-l-2-term"]], "Exercise: Robust preconditioners": [[104, "exercise-robust-preconditioners"]], "Exercise A:": [[104, "exercise-a"]], "Exercise B:": [[104, "exercise-b"]], "Domain Decomposition with minimal overlap": [[105, "domain-decomposition-with-minimal-overlap"]], "Analysis of the method": [[105, "analysis-of-the-method"]], "Adding a coarse grid space": [[105, "adding-a-coarse-grid-space"]], "Graph-based mesh partitioning": [[105, "graph-based-mesh-partitioning"]], "Overlapping Domain Decomposition Methods": [[106, "overlapping-domain-decomposition-methods"]], "Experiments with overlapping DD": [[106, "experiments-with-overlapping-dd"]], "Analysis of the DD preconditioner": [[106, "analysis-of-the-dd-preconditioner"]], "Overlapping DD Methods with coarse grid": [[106, "overlapping-dd-methods-with-coarse-grid"]], "Analysis of the 2-level method:": [[106, "analysis-of-the-2-level-method"]], "Comparison to DD with minimal overlap": [[106, "comparison-to-dd-with-minimal-overlap"]], "Schwarz preconditioners for high order finite elements": [[107, "schwarz-preconditioners-for-high-order-finite-elements"]], "Overlapping blocks": [[107, "overlapping-blocks"]], "Runge Kutta Methods": [[108, "runge-kutta-methods"]], "Butcher tableaus of simple methods": [[108, "butcher-tableaus-of-simple-methods"]], "Convergence of Runge Kutta methods": [[108, "convergence-of-runge-kutta-methods"]], "Explicit methods:": [[108, "explicit-methods"]], "Diagonally implicit Runge-Kutta methods:": [[108, "diagonally-implicit-runge-kutta-methods"]], "Exercise:": [[108, "exercise"], [118, "exercise"]], "Single-step methods": [[109, "single-step-methods"]], "Simple methods based on the integral equation": [[109, "simple-methods-based-on-the-integral-equation"]], "Explicit Euler method (EE)": [[109, "explicit-euler-method-ee"]], "Implicit Euler method (IE)": [[109, "implicit-euler-method-ie"]], "Trapezoidal method": [[109, "trapezoidal-method"]], "Explicit mid-point rule (=improved Euler method = RK2 method)": [[109, "explicit-mid-point-rule-improved-euler-method-rk2-method"]], "Linear stability classification": [[109, "linear-stability-classification"]], "Stability function": [[109, "stability-function"]], "Stability conditions": [[109, "stability-conditions"]], "Single step methods and parabolic equations": [[109, "single-step-methods-and-parabolic-equations"]], "Single step methods and hyperbolic equations": [[109, "single-step-methods-and-hyperbolic-equations"]], "Exercises:": [[109, "exercises"]], "Heat Equation": [[110, "heat-equation"]], "Variational formulation in space": [[110, "variational-formulation-in-space"]], "Galerkin method in space": [[110, "galerkin-method-in-space"]], "Implicit Euler time-stepping": [[110, "implicit-euler-time-stepping"]], "Wave Equation": [[111, "wave-equation"]], "Newmark time-stepping method": [[111, "newmark-time-stepping-method"]], "Exponential Integrators for Parabolic Equations": [[112, "exponential-integrators-for-parabolic-equations"]], "Mass-lumping and Local time-stepping": [[115, "mass-lumping-and-local-time-stepping"]], "The Verlet method:": [[115, "the-verlet-method"]], "Geometry with local details:": [[115, "geometry-with-local-details"]], "Nano-optics: A ring-resonator": [[116, "nano-optics-a-ring-resonator"], [117, "nano-optics-a-ring-resonator"]], "Discontinuous Galerkin for the Wave Equation": [[118, "discontinuous-galerkin-for-the-wave-equation"]], "Testing the differential operators": [[118, "testing-the-differential-operators"]], "Efficient implementation:": [[118, "efficient-implementation"]], "Solving the wave equation:": [[118, "solving-the-wave-equation"]], "Eigenvalues of the discretized Laplace-operator": [[118, "eigenvalues-of-the-discretized-laplace-operator"]], "Equivalent norms on H^1 and on sub-spaces": [[98, "equivalent-norms-on-h-1-and-on-sub-spaces"]], "Generalized derivatives": [[95, "generalized-derivatives"]], "Friedrichs\u2019 inequality": [[99, "friedrichs-inequality"], [100, "friedrichs-inequality"]], "Poincar\u00e9 inequality": [[99, "poincare-inequality"], [100, "poincare-inequality"]], "Friedrichs\u2019 inequality in 1D": [[99, "friedrichs-inequality-in-1d"]], "Friedrichs\u2019 inequality on the square": [[99, "friedrichs-inequality-on-the-square"]], "Poincar\u00e9 inequality in 1D": [[99, "poincare-inequality-in-1d"]], "Bramble-Hilbert Lemma": [[99, "bramble-hilbert-lemma"]], "Mean-value interpolation": [[99, "mean-value-interpolation"]], "Scaled domain": [[99, "scaled-domain"]], "Fractional Sobolev spaces": [[99, "fractional-sobolev-spaces"]], "Step function": [[99, "step-function"]], "Point evaluation functional": [[99, "point-evaluation-functional"]], "Sobolev spaces": [[96, "sobolev-spaces"]], "The weak formulation of the Poisson equation": [[101, "the-weak-formulation-of-the-poisson-equation"]], "Shift theorems": [[101, "shift-theorems"]], "Experiments with norms": [[100, "experiments-with-norms"]], "Inverse estimates": [[100, "inverse-estimates"]], "Trace inequality": [[100, "trace-inequality"]], "Korn\u2019s inequality": [[100, "korn-s-inequality"]], "Non-linear dynamics": [[57, "non-linear-dynamics"]], "3D-TDNNS Elasticity and Reissner Mindlin Plate elements": [[69, "d-tdnns-elasticity-and-reissner-mindlin-plate-elements"]], "Trace theorems and their applications": [[97, "trace-theorems-and-their-applications"]], "Integration by parts": [[97, "integration-by-parts"]], "Sobolev spaces over sub-domains": [[97, "sobolev-spaces-over-sub-domains"]], "Extension operators": [[97, "extension-operators"]], "The trace space H^{1/2}": [[97, "the-trace-space-h-1-2"]], "Interpolation spaces": [[97, "interpolation-spaces"]], "General definition:": [[97, "general-definition"]], "The trace space on one edge of the square": [[97, "the-trace-space-on-one-edge-of-the-square"]]}, "indexentries": {}}) \ No newline at end of file diff --git a/sobolevspaces/Traces.html b/sobolevspaces/Traces.html index c538fedd..72425897 100644 --- a/sobolevspaces/Traces.html +++ b/sobolevspaces/Traces.html @@ -203,7 +203,7 @@
  • 15. Trace theorems and their applications
  • 16. Equivalent norms on \(H^1\) and on sub-spaces
  • 17. The weak formulation of the Poisson equation
    • -
  • 18. Experiments with norms (WIP)
    • +
  • 18. Experiments with norms
  • 19. Exercises

    • Finite Element Method

    @@ -675,6 +675,7 @@

    15.2. Sobolev spaces over sub-domains

    The interfaces are \(\gamma_{ij} = \overline \Omega_i \cap \overline \Omega_j\). The outer normal vector of \(\Omega_i\) is \(n_i\).

    +    Alternative text

    Theorem: Let \(u \in L_2(\Omega)\) such that

    @@ -764,8 +765,8 @@

    15.3. Extension operators\(\widetilde \Omega = \Omega \cup S_1 \cup \ldots \cup S_M\).

    We define the extension operator by

    -
    -()#\[\begin{equation} +
    +()#\[\begin{equation} \begin{array}{rcll} (E u) (\hat x) & = & u(x) & \qquad \forall \, x \in \cup S_i \\ (Eu) (x) & = & u(x) & \qquad \forall \, x \in \Omega @@ -1000,8 +1001,8 @@

    15.5.1. General definition:\((z_k^\prime)^\prime = (1-\lambda_k) z_k\), i.e., \(z^k \in H^2\). Since \(H^2 \subset C^0\), there holds also \(z \in C^2\), and a weak solution is also a solution of the strong form

    -
    -()#\[\begin{equation} +
    +()#\[\begin{equation} \begin{array}{rcll} z_k - z_k^{\prime \prime} & = & \lambda_k z_k \qquad & \mbox{ on } (0,1) \\ z_k^\prime(0) = z_k^\prime(1) & = & 0