Multilevel Finite Element Approximation: Theory and Applications

ISBN-13: 9783519027195 / Angielski / Miękka / 1994 / 160 str.

Peter Oswald; Peter Oswald

Opis

These notes reflect, to a great part, the present research interests of the author but were influenced by the ideas and the work of many colleagues. They are based on lectures given by the author at the Institutes of Mathematics and Informatics at the Technical U niversity of Munich during February /March 1993. I wish to warmly thank ehr. Zenger and R. Hoppe for their generous support and the many discussions I had with them and their younger colleagues during the last year. Part of the results contained in section 4 is the output of these discussions and joint work with M. Griebel. There are many other mathematicians who encouraged me (or personally or by their mathematical work) to step into the field of multilevel methods. I want to acknowledge the support I received from W. Dahmen, R. A. DeVore, P. Deufl hard, W. Hackbusch, H. Trieb el, O. Widlund, H. Yserentant and many others. On the other hand, I should apologize for not mentioning many interesting re search results and names standing for recent developments in the fields which are the subject of these notes. Finally, I want to thank my family, my wife Olga and my daughters Evelyn and Annelie, for their everyday patience and support.

Spis treści

1 Introduction.- 2 Finite element approximation.- 2.1 Finite elements, multivariate splines, wavelets.- 2.1.1 Finite element subspaces.- 2.1.2 Spline spaces.- 2.1.3 Wavelets.- 2.2 Moduli of smoothness and K-functionals.- 2.3 Jackson and Whitney inequalities.- 2.4 Bernstein inequalities and inverse estimates.- 2.5 Information on other approximation schemes.- 2.6 Constructive characterization of Besov spaces.- 3 Function spaces.- 3.1 Spaces on Rd.- 3.1.1 Fourier decomposition methods.- 3.1.2 Other techniques and spaces.- 3.2 Spaces on domains and extension.- 3.3 Spaces on manifolds and traces.- 3.4 Approximation spaces on polyhedral domains.- 3.4.1 Definition and general properties.- 3.4.2 Approximation theory in the Ap,qs scale.- 3.4.3 Norms on Vj and special representations.- 3.4.4 Extensions, traces, boundary conditions.- 3.4.5 Output: Decomposition norms in Hs.- 4 Applications to multilevel methods.- 4.1 The abstract Schwarz theory.- 4.2 Second-order elliptic equations.- 4.2.1 The basic multilevel preconditioners.- 4.2.2 Nested refinement.- 4.2.3 Further developments and problems.- 4.3 The biharmonic problem.- 4.4 Domain decomposition and boundary element methods.- 4.5 Sparse grids.- 4.6 Nonconforming and mixed methods.- 4.6.1 Splittings for nonconforming elements.- 4.6.2 Mixed finite element methods.- 5 Error estimates and adaptivity.- 5.1 Traditional error estimates.- 5.2 h-version and nonlinear approximation.- 5.3 Adaptive multilevel methods.- 5.4 More complicated approximation schemes.- 5.4.1 The h-p-version.- 5.4.2 Wavelet packets and compression.- 5.4.3 Approximation with long rectangles.- References.