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Embedding and a priori wavelet-adaptivity for Dirichlet problems. (English) Zbl 0985.65149
Let \(\Omega\subset R^d\) be a bounded domain with Lipschitz continuous boundary embedded in a rectangular domain \(\Theta\). The Dirichlet boundary value problem in \(\Omega\) is extended to a periodic boundary value problem over \(\Theta\). For the corresponding variational problem (called the fictitious domain formulation), a Galerkin discretization is applied with subspaces generated by integer translations and binary dilations of a single biorthogonal compactly supported scaling function. Such a domain embedding method was proposed and analyzed in the author’s previous paper [RAIRO, Modélisation Math. Anal. Numér. 32, No. 4, 405-431 (1998; Zbl 0913.65099)].
Now, in order to improve accuracy, the author proposes an a priori adaptive strategy using a finer discretization near the boundary of \(\Omega\). This is done by a modification of approximation subspaces via selecting suitable wavelet subspaces. The theoretical result is illustrated by numerical experiments.

65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65T60 Numerical methods for wavelets
65N50 Mesh generation, refinement, and adaptive methods for boundary value problems involving PDEs
35J25 Boundary value problems for second-order elliptic equations
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