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The approximation and computation of a basis of the trace space \(H^{1/2}\). (English) Zbl 1122.65119

A method for the construction of an approximate basis of the trace space based on a combination of the Steklov spectral method and a finite element approximation is presented. The Steklov eigenfunctions are approximated with respect to a particular finite element basis. Then solutions of elliptic boundary value problems with Dirichlet boundary conditions can be efficiently and accurately expanded in the discrete Steklov basis. A reformulation of the discrete Steklov eigenproblem is solved by the implicitly restarted Arnoldi method ARPACK.
Examples for the solution of elliptic problems on bounded domains using both a nonconforming bilinear finite element and a non-conforming harmonic finite element method are included. In addition, the efficiency of the proposed method is documented for the Laplace equation on a densely perforated domain.

MSC:

65N35 Spectral, collocation and related methods for boundary value problems involving PDEs
35J25 Boundary value problems for second-order elliptic equations
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs

Software:

IRAM; ARPACK; eigs
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Full Text: DOI

References:

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