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Spectral element discretizations of the Poisson equation with mixed boundary conditions. (English) Zbl 1004.65119
The paper presents optimal error estimations for the Poisson equation with boundary conditions of mixed Dirichlet and Neumann types in a unit square using a spectral element discretization. Starting with a variational formulation of the problem a lack of regularity of the solution is observed. This determines the convergence order of the best polynomial approximation. Using a pure spectral method, i.e. without domain decomposition, an optimal error estimate is proven, where the order of convergence is explicitly given. Introducing a conforming domain decomposition of the square into four rectangles spectral elements allow to improve the accuracy of the discretization. Therefore polynomials of different degrees are used on the different subdomains. An error analysis yields the optimal order of convergence of the spectral element discretization.

65N35 Spectral, collocation and related methods for boundary value problems involving PDEs
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
65N55 Multigrid methods; domain decomposition for boundary value problems involving PDEs
65N15 Error bounds for boundary value problems involving PDEs
65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs