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Diagonalized Legendre spectral methods using Sobolev orthogonal polynomials for elliptic boundary value problems. (English) Zbl 1382.65424

Summary: Fully diagonalized spectral methods using Sobolev orthogonal/biorthogonal basis functions are proposed for solving second-order elliptic boundary value problems. We first construct the Fourier-like Sobolev polynomials which are mutually orthogonal (resp. bi-orthogonal) with respect to the bilinear form of the symmetric (resp. unsymmetric) elliptic Neumann boundary value problems. The exact and approximation solutions are then expanded in an infinite and truncated series in the Sobolev orthogonal polynomials, respectively. An identity is also established for the a posterior error estimate with a simple error indicator. Further, the Fourier-like Sobolev orthogonal polynomials and the corresponding Legendre spectral method are proposed in parallel for Dirichlet boundary value problems. Numerical experiments illustrate that our Legendre methods proposed are not only efficient for solving elliptic problems but also equally applicable to indefinite Helmholtz equations and singular perturbation problems.

MSC:

65N35 Spectral, collocation and related methods for boundary value problems involving PDEs
35J25 Boundary value problems for second-order elliptic equations
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
35B25 Singular perturbations in context of PDEs
65N15 Error bounds for boundary value problems involving PDEs
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