Solving Minimal Residual Methods in $W^{-1,p'}$ with large Exponents $p$

TL;DR

This paper presents a numerical scheme for solving linear PDEs by minimizing residuals in a specialized Sobolev space norm with large exponents, improving stability for convection-dominated problems.

Contribution

It introduces a novel regularized Kacanov iteration method to efficiently handle non-linear minimization in high exponent regimes for PDE solutions.

Findings

Effective approximation of PDE solutions in $W^{-1,p'}$ norm for large $p$

Enhanced stability in convection-dominated diffusion problems

Applicable to non-linear residual minimization schemes

Abstract

We introduce a numerical scheme that approximates solutions to linear PDE's by minimizing a residual in the $W^{-1,p'}(\Omega)$ norm with exponents $p> 2$. The resulting problem is solved by regularized Kacanov iterations, allowing to compute the solution to the non-linear minimization problem even for large exponents $p\gg 2$. Such large exponents remedy instabilities of finite element methods for problems like convection-dominated diffusion.