Numerical analysis for the Stokes problem with non-homogeneous Dirichlet boundary condition

TL;DR

This paper develops and analyzes numerical methods for solving the Stokes problem with non-homogeneous boundary conditions, including cases with low regularity and corner singularities, providing error estimates and numerical validation.

Contribution

It introduces new error estimates for conforming discretizations of the Stokes problem with complex boundary data and studies the well-posedness of very weak solutions.

Findings

Optimal discretization error estimates are derived.

Numerical tests confirm the theoretical results.

The theory accounts for corner singularities and low regularity boundary data.

Abstract

The Stokes problem with non-homogeneous Dirichlet boundary condition is solved numerically using conforming discretizations and an approximation of the boundary datum in the corresponding trace space. Optimal discretization error estimates are derived. The theory accounts for the influence of corner singularities in the case of a non-convex domain. Several variants of the boundary data approximation are discussed. Moreover, the case of boundary data with very low regularity is studied, where a weak solution does not exist. The well-posedness of the very weak solution is investigated, and optimal discretization error estimates are derived. Numerical tests confirm the theory. The compatibility condition for the boundary data is not necessary for well-posedness of the weak and very weak formulations but it ensures that the solution satisfies the continuity equation in the distributional sense. In the same spirit, the compatibility condition is not necessary for the approximating boundary data; a good approximation of the original boundary data is important.