| Preface 1 Introduction 2 Evolutionary Variational Inequality Approach 2.1 The degenerate free boundary problem 2.2 Some application problems 2.3 Different fixed domain formulations 2.3.1 Front tracking and fixing methods versus fixed domain formulations exemplified by injection and compression moulding 2.3.2 Weak formulation 2.3.3 The evolutionary variational inequality approach 3 Properties of the Variational Inequality Solution 3.1 Problem setting and general notations 3.2 Existence and uniqueness result 3.3 Monotonicity properties and regularity with respect to time 3.3.1 Time-independent convex sets 3.3.2 Time-dependent convex sets 3.4 Regularity with respect to space variables 3.4.1 Dirichlet boundary conditions 3.4.2 Boundary conditions of Neumann/Newton type 3.5 Some remarks on further regularity results 4 Finite Volume Approximations for Elliptic Inequalities 4.1 Finite element and volume approximations forthe obstacle problem 4.1.1 The elliptic obstacle problem 4.1.2 Finite element approximations for the obstacle problem 4.1.3 Basics of finite volume approximations 4.1.4 Finite volume approximations for the obstacle problem 4.2 Comparison of finite volume and finite element approximations 4.3 Error estimates for the finite volume solution 4.4 Penalization methods for the finite volume obstacle problem . 4.4.1 Discrete maximum principle 4.4.2 Discussion of penalization techniques 4.4.3 Iterative solution of the penalization problems 4.5 The Signorini problem as a boundary obstacle problem 4.6 Results from numerical experiments for elliptic obstacle problems 4.6.1 Examples with known exact solution 4.6.2 Numerical results for the error between the finite element and the finite volume solution 4.6.3 Error behaviour of the finite volume and the penalization solutions 5 Numerical Analysis of the Evolutionary Inequalities 5.1 Finite element and volume approximations for the evolutionary problems 5.1.1 Formulation of the finite element and finite volume approximations 5.1.2 Properties of the discrete inequality problems 5.1.3 Time evolution of the finite volume solution 5.2 Error estimates for the finite element and finite volume solutions 5 2 1 Cnmp~rison of the finite clemcnt and finitc volume approximations 5.2.2 A priori estimates for the finite element and finite volume solutions 5.2.3 Convergence rate for the finite element and finite volume solutions 5.3 Penalization methods for the evolutionary finite volume inequalities 5.3.1 Discussion of penalization techniques 5.3.2 Iterative solution of the penalization problems 5.4 Numerical experiments for evolutionary variational inequalities 5.4.1 Two evolutionary variational inequalities and the related free boundary problems 5.4.2 Numerical results for the errors between exact, finite element and finite volume solution 5.4.3 Error behaviour of the penalization solutions …… 6 Injection and Compression Mouding as Application problems 7 Concluding remarks Bibliography List of Figures List of Tables List of Symbols Index |
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