Wavelet Least Squares Methods for Boundary Value Problems

This paper is concerned with least squares methods for the numerical solution of operator equations. Our primary focus is the discussion of the following conceptual issues: the selection of appropriate least squares functionals, their numerical evaluation in the special light of recent developments...

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Bibliographic Details
Published in:SIAM journal on numerical analysis 2002, Vol.39 (6), p.1985-2013
Main Authors: Dahmen, Wolfgang, Angela Kunoth, Schneider, Reinhold
Format: Article
Language:English
Subjects:
Approximation
Boundary conditions
Boundary value problems
Exact sciences and technology
Index sets
Inner products
Lagrange multipliers
Least squares
Mathematics
Matrices
Norms
Numerical analysis
Numerical analysis in abstract spaces
Numerical analysis. Scientific computation
Numerical linear algebra
Partial differential equations, boundary value problems
Sciences and techniques of general use
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Summary:This paper is concerned with least squares methods for the numerical solution of operator equations. Our primary focus is the discussion of the following conceptual issues: the selection of appropriate least squares functionals, their numerical evaluation in the special light of recent developments of wavelet methods, and a natural way of preconditioning the resulting systems of linear equations. We describe first a general format of variational problems that are well-posed in a certain natural topology. In order to illustrate the scope of these problems we identify several special cases such as second order elliptic boundary value problems, their formulation as a first order system, transmission problems, the system of Stokes equations, or more general saddle point problems. Particular emphasis is placed on the separate treatment of essential nonhomogeneous boundary conditions. We propose a unified treatment based on wavelet expansions. In particular, we exploit the fact that weighted sequence norms of wavelet coefficients are equivalent to relevant function norms arising in the least squares context. This provides access to "difficult" norms, efficient preconditioners and, in the case of first order systems, optimal L2error estimates.
ISSN:0036-1429
1095-7170
DOI:10.1137/S0036142999361852