Perturbed saddle-point problems in $\mathbf{L}^p$ with non-regular loads

TL;DR

This paper develops a new analysis framework for perturbed saddle-point problems in Banach spaces, focusing on non-regular loads, and demonstrates its effectiveness through theoretical results and numerical experiments.

Contribution

It introduces a novel discrete solvability analysis for perturbed saddle-point problems with non-regular loads, including a supercloseness result and convergence analysis of a postprocessing scheme.

Findings

The proposed scheme converges for loads in $ ext{H}^{-1}$.

Supercloseness results improve solution accuracy.

Numerical results confirm theoretical convergence.

Abstract

In this work, we develop the discrete solvability analysis for perturbed saddle-point problems in Banach spaces with forcing terms regularised by means of a projector constructed using the adjoint of a weighted Cl\'ement quasi-interpolation. We take as driving example the linearised Poisson--Boltzmann (an advection-diffusion-reaction problem) in mixed form. We use perturbation arguments on the continuous and discrete levels and then derive a priori estimates that remain valid when the load that appears on the right-hand side of the "second" equation is in $\mathrm{H}^{-1}$. Further, we show a supercloseness result and {analyse convergence} of an adequate adaptation of Stenberg postprocessing for mixed advection equations with non-regular data. We provide numerical results that illustrate the convergence of the proposed scheme.