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

Abstract

We introduce a numerical scheme that approximates solutions to linear PDE's by minimizing a residual in the $W^{-1,p^{\prime }}(\Omega )$ 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.