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Finite-element trace theorems. (English) Zbl 0960.65129

The author proves the finite-element trace theorem for the Sobolev spaces \(H^1_{p,q}\) and then studies the properties of Poincaré-Steklov operators for anisotropic elliptic problems.
The main aim of the article is to study the space of the traces of the finite-element functions on the boundaries of a bounded polygonal domain \(\Omega\) which is generated by the \(H_{p,q}^1(\Omega)\) norm. The author proves that the corresponding constants in the trace theorems are independent of the parameters. Sobolev spaces with parameter-dependent norms are generated, for instance, by elliptic problems with disproportional anisotropic coefficients. These results are used for obtaining effective domain decomposition methods.

MSC:

65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
65N55 Multigrid methods; domain decomposition for boundary value problems involving PDEs
35J25 Boundary value problems for second-order elliptic equations
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