Note for Advance Numerical Methods - ANM By Abhishek Apoorv

Topics ( 8 )

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Advanced Numerical Methods and Their Applications to Industrial Problems — Adaptive Finite Element Methods Lecture Notes Summer School Yerevan State University Yerevan, Armenia 2004 Alfred Schmidt, Arsen Narimanyan Center for Industrial Mathematics University of Bremen Bremen, Germany www.math.uni-bremen.de/zetem/

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Contents 1 Introduction, motivation 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Multiple scales in the modelling of real world problems . . . . . . . . . . 1 1 2 2 Mathematical modeling 2.1 Density, flux, and conservation . . . . . . . . . . . . . . . . . . . . . . . . 2.2 PDEs as a modeling tool . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 5 11 3 Functional analysis background 3.1 Banach spaces and Hilbert spaces . . . . 3.2 Basic concepts of Lebesgue spaces . . . . 3.3 Weak derivatives . . . . . . . . . . . . . 3.4 Introduction to Sobolev spaces . . . . . . 3.5 Some useful properties of Sobolev spaces . . . . . 15 15 16 17 18 19 4 Variational formulation of elliptic problems 4.1 Variational formulation of Poisson problem . . . . . . . . . . . . . . . . . 4.2 Existence and uniqueness of weak solution . . . . . . . . . . . . . . . . . 21 21 22 5 Finite element approximation 5.1 Galerkin discretization . . . . . . . 5.2 Finite element method . . . . . . . 5.3 Discretisation of 2nd order equation 5.4 Simplices of arbitrary dimension . . . . . . 25 25 26 31 34 . . . . . 37 37 38 39 42 45 7 A posteriori error estimation for elliptic problems 7.1 A posteriori error estimation in the energy norm . . . . . . . . . . . . . . 46 46 8 Mesh refinement and coarsening 8.1 Refinement algorithms for simplicial meshes 8.2 Prolongation of data during refinement . . . 8.3 Coarsening algorithms . . . . . . . . . . . . 8.4 Restriction of data during coarsening . . . . 8.5 Storage methods for hierarchical meshes . . 51 52 58 58 60 61 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 A priori error estimates for elliptic problems 6.1 Abstract error estimates: C´ea’s lemma . . . . 6.2 Interpolation estimates . . . . . . . . . . . . . 6.2.1 Clement interpolation . . . . . . . . . 6.2.2 Lagrange interpolation . . . . . . . . . 6.3 A priori error estimate . . . . . . . . . . . . . i . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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9 Adaptive strategies for elliptic problems 9.1 Quasi-optimal meshes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2 Mesh refinement strategies . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3 Coarsening strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 63 64 68 10 Aspects of efficient implementation 10.1 Numerical integration (quadrature schemes) . . . . . . . . . . . . . . . . 10.2 Efficient solvers for linear systems . . . . . . . . . . . . . . . . . . . . . . 10.2.1 Methods of Jacobi, Gauss-Seidel and Relaxation . . . . . . . . . . 10.2.2 Conjugate gradient method and other Krylov subspace iterations 10.2.3 Multilevel preconditioners and solvers . . . . . . . . . . . . . . . . 70 70 71 71 75 76 11 Error estimates via dual techniques 11.1 Estimates for the L2 norm of the error . . . . . . . . . . . . . . . . . . . 11.2 A priori error estimation in the L2 norm . . . . . . . . . . . . . . . . . . 11.3 A posteriori error estimation in the L2 norm . . . . . . . . . . . . . . . . 77 77 78 78 12 Parabolic problems - heat equation 12.1 Weak solutions of heat equation . . . . . . . . . . . . . . . . . . . . . . . 12.2 Discretization of heat equation . . . . . . . . . . . . . . . . . . . . . . . . 12.3 A priori error estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 81 83 86 13 A posteriori error estimates for parabolic problems 13.1 Abstract error estimate for ordinary differential equations 13.2 Weak formulation of the heat equation . . . . . . . . . . 13.3 Discretization . . . . . . . . . . . . . . . . . . . . . . . . 13.4 Error representation and dual problem . . . . . . . . . . 13.5 A posteriori error estimate . . . . . . . . . . . . . . . . . . . . . . 87 87 89 89 90 92 14 Adaptive methods for parabolic problems 14.1 Adaptive control of the time step size . . . . . . . . . . . . . . . . . . . . 93 94 15 The 15.1 15.2 15.3 15.4 15.5 15.6 15.7 Stefan problem of phase transition Problem setting: . . . . . . . . . . . . . Discretization . . . . . . . . . . . . . . . Error control for Stefan problem . . . . . Error Representation Formula . . . . . . A Posteriori Error Estimators . . . . . . Adaptive Algorithm . . . . . . . . . . . . Numerical Experiments . . . . . . . . . . 15.7.1 Example 1: Oscillating Circle . . 15.7.2 Example 2: Oscillating Source . . 15.8 Nonlinear solver . . . . . . . . . . . . . . ii . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 99 99 100 100 101 102 103 103 103 106

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16 The continuous casting problem 16.1 Setting . . . . . . . . . . . . . . 16.2 Discretization . . . . . . . . . . 16.3 Parabolic Duality . . . . . . . . 16.4 Robin Inflow Condition . . . . . 16.5 Dirichlet Inflow Condition . . . 16.6 Discontinuous p . . . . . . . . . 16.7 Error Representation Formula . 16.8 A Posteriori Error Analysis . . 16.9 Residuals . . . . . . . . . . . . 16.10Proof of Theorem 16.1 . . . . . 16.11Proof of Theorem 16.2 . . . . . 16.12Discontinuous p . . . . . . . . . 16.13Performance . . . . . . . . . . . 16.14Localization and adaption . . . 16.15Example: Traveling wave . . . . 16.16Applications to Casting of Steel 16.17Scaling . . . . . . . . . . . . . . 16.18Example: Oscillating Velocity . 16.19Example: Oscillating Cooling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 Mathematical modeling of thermal cutting 17.1 Introduction . . . . . . . . . . . . . . . . . . . . . . 17.2 Problem description and physical modeling . . . . . 17.3 Mathematical formulation of the problem . . . . . . 17.4 Variational inequalities and the weak formulation of 17.4.1 Notation and Functional Spaces . . . . . . . 17.4.2 A VI equivalent of Stefan-Signorini problem 17.5 Level set formulation . . . . . . . . . . . . . . . . . 17.5.1 Stefan condition as level-set equation . . . . 17.6 Weak formulation of Stefan-Signorini problem . . . 17.7 Heat Flux Density . . . . . . . . . . . . . . . . . . 17.8 Solution algorithm . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . the . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 109 111 113 114 115 116 117 118 121 125 125 125 126 126 127 129 129 132 133 . . . . . . . . . . . . . . . problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 135 135 137 139 139 140 141 141 144 144 148 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150 iii