A Petrov-Galerkin multilayer discretization to second order elliptic boundary value problems
Provides a reliable discretization framework for shallow fluid flow problems with diffusion, but the contribution is incremental as it extends existing methods to a multilayer setting.
The paper develops a Petrov-Galerkin multilayer discretization for second-order elliptic problems, proving well-posedness and optimal error estimates. Numerical tests on smooth problems confirm the theoretical results.
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 trial 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.