Multilevel Finite Element Approximation Theory and Applications [Paperback]

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.Springer Book Archives