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Since the early 1960s, the mathematical theory of variational inequalities has been under rapid development, based on complex analysis and strongly influenced by 'real-life' application. Many, but of course not all, moving free (Le. , a priori un known) boundary problems originating from engineering and economic applica tions can directly, or after a transformation, be formulated as variational inequal ities. In this work we investigate an evolutionary variational inequality with a memory term which is, as a fixed domain formulation, the result of the application of such a transformation to a degenerate moving free boundary problem. This study includes mathematical modelling, existence, uniqueness and regularity results, numerical analysis of finite element and finite volume approximations, as well as numerical simulation results for applications in polymer processing. Essential parts of these research notes were developed during my work at the Chair of Applied Mathematics (LAM) of the Technical University Munich. I would like to express my sincerest gratitude to K. -H. Hoffmann, the head of this chair and the present scientific director of the Center of Advanced European Studies and Research (caesar), for his encouragement and support. With this work I am fol lowing a general concept of Applied Mathematics to which he directed my interest and which, based on application problems, comprises mathematical modelling, mathematical and numerical analysis, computational aspects and visualization of simulation results.
List of contents
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.- 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.- 3.4 Regularity with respect to space variables.- 3.5 Some remarks on further regularity results.- 4 Finite Volume Approximations for Elliptic Inequalities.- 4.1 Finite element and 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.- 4.5 The Signorini problem as a boundary obstacle problem.- 4.6 Results from numerical experiments for elliptic obstacle problems.- 5 Numerical Analysis of the Evolutionary Inequalities.- 5.1 Finite element and volume approximations for the evolutionary problems.- 5.2 Error estimates for the finite element and finite volume solutions.- 5.3 Penalization methods for the evolutionary finite volume inequalities.- 5.4 Numerical experiments for evolutionary variational inequalities.- 6 Injection and Compression Moulding as Application Problems.- 6.1 Classical Hele-Shaw flows and related moving boundary problems.- 6.2 Mathematical modelling of injection and compression moulding.- 6.3 Simulation results.- 7 Concluding Remarks.- List of Figures.- List of Tables.- List of Symbols.
Summary
Since the early 1960s, the mathematical theory of variational inequalities has been under rapid development, based on complex analysis and strongly influenced by 'real-life' application. Many, but of course not all, moving free (Le. , a priori un known) boundary problems originating from engineering and economic applica tions can directly, or after a transformation, be formulated as variational inequal ities. In this work we investigate an evolutionary variational inequality with a memory term which is, as a fixed domain formulation, the result of the application of such a transformation to a degenerate moving free boundary problem. This study includes mathematical modelling, existence, uniqueness and regularity results, numerical analysis of finite element and finite volume approximations, as well as numerical simulation results for applications in polymer processing. Essential parts of these research notes were developed during my work at the Chair of Applied Mathematics (LAM) of the Technical University Munich. I would like to express my sincerest gratitude to K. -H. Hoffmann, the head of this chair and the present scientific director of the Center of Advanced European Studies and Research (caesar), for his encouragement and support. With this work I am fol lowing a general concept of Applied Mathematics to which he directed my interest and which, based on application problems, comprises mathematical modelling, mathematical and numerical analysis, computational aspects and visualization of simulation results.
