A Petrov-Galerkin multilayer discretization to second order elliptic boundary value problems. (English) Zbl 1442.65350
Summary: We study in this paper a multilayer discretization of second order elliptic problems, aimed at providing reliable multilayer discretizations of shallow fluid flow problems with diffusive effects. This discretization is based upon the formulation by transposition of the equations. It is a Petrov-Galerkin discretization in which the trial functions are piecewise constant per horizontal layers, while the test functions are continuous piecewise linear, on a vertically shifted grid.
We prove the well posedness and optimal error order estimates for this discretization in natural norms, based upon specific inf-sup conditions.
We present some numerical tests with parallel computing of the solution based upon the multilayer structure of the discretization, for academic problems with smooth solutions, with results in full agreement with the theory developed.
We prove the well posedness and optimal error order estimates for this discretization in natural norms, based upon specific inf-sup conditions.
We present some numerical tests with parallel computing of the solution based upon the multilayer structure of the discretization, for academic problems with smooth solutions, with results in full agreement with the theory developed.
MSC:
| 65N30 | Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs |
| 35J25 | Boundary value problems for second-order elliptic equations |
Keywords:
multilayer methods; transposition solution; finite differences; Petrov-Galerkin discretizationsReferences:
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