A Petrov-Galerkin multilayer discretization to second order elliptic boundary value problems

TL;DR

This paper introduces a Petrov-Galerkin multilayer discretization method for second order elliptic boundary value problems, suitable for shallow fluid flow simulations with diffusive effects, demonstrating well-posedness and optimal error estimates.

Contribution

It presents a novel multilayer Petrov-Galerkin discretization with proven stability and accuracy for elliptic problems, tailored for shallow fluid flow applications.

Findings

Numerical tests confirm theoretical error estimates.

Discretization is well-posed with optimal error bounds.

Parallel computing effectively solves academic problems.

Abstract

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.