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

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