Minimum residual discretization of a semilinear elliptic problem
Carlos García Vera, Norbert Heuer, and Dirk Praetorius
We propose a least-squares penalization as a means to extend the
discontinuous Petrov–Galerkin (DPG) method with optimal test functions
to a class of semilinear elliptic problems.
The nonlinear contributions are replaced with independent
unknowns so that standard DPG techniques apply to the then linear problem
with non-trivial kernel. The nonlinear relations are added as
least-squares constraints. Assuming solvability of the semilinear problem
and an Aubin–Nitsche-type approximation property for the primal variable, we prove
a Cáa estimate for the approximation error in canonical norms.
Numerical results with uniform and adaptively refined meshes
illustrate the performance of the scheme.