An a posteriori error analysis for a coupled continuum pipe-flow/Darcy model in Karst aquifers: anisotropic and isotropic discretizations

TL;DR

This paper develops an a posteriori error analysis for a coupled pipe-flow/Darcy model in karst aquifers, accommodating anisotropic and isotropic finite element discretizations, with bounds that are robust to mesh anisotropy.

Contribution

It provides the first unified error bounds for anisotropic finite element discretizations of coupled flow models in karst aquifers, including both conforming and nonconforming elements.

Findings

Lower error bounds are uniform regardless of mesh anisotropy.

Upper error bounds depend on mesh alignment with anisotropy.

Results simplify for isotropic meshes, providing unconditional bounds.

Abstract

This paper presents an a posteriori error analysis for a coupled continuum pipe-flow/Darcy model in karst aquifers. We consider a unified anisotropic finite element discretization (i.e. elements with very large aspect ratio). Our analysis covers two-dimensional domains, conforming and nonconforming discretizations as well as different elements. Many examples of finite elements that are covered by analysis are presented. From the finite element solution, the error estimators are constructed and based on the residual of model equations. Lower and upper error bounds form the main result with minimal assumptions on the elements. The lower error bound is uniform with respect to the mesh anisotropy in the entire domain. The upper error bound depends on a proper alignment of the anisotropy of the mesh which is a common feature of anisotropic error estimation. In the special case of isotropic meshes, the results simplify, and upper and lower error bounds hold unconditionally.