The approximation and computation of a basis of the trace space \(H^{1/2}\) (Q2641369)

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.