  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.
