On the convergence of solutions for a class of nonconformal FEM schemes for quasilinear elliptic equations (Q2434668)

The author considers a quasilinear elliptic equation in a polygonal domain in \(R^n\) along with Dirichlet boundary conditions and assuming Lipschitz-continuous nonlinearities. For this problem he proposes a nonconforming finite element method (FEM) on a uniformly regular triangulation using elementwise linear functions, introducing, on the faces, control vectors and requiring the normals to the faces to satisfy a condition connected to these control vectors. His bilinear form does not contain penalty terms but integrals containing the jumps of the test functions. Proving in the two-dimensional case a Poincaré-Friedrichs-like inequality and referring to \textit{S. C. Brenner} [SIAM J. Numer. Anal. 41, No. 1, 306--324 (2003; Zbl 1045.65100)], he shows stability and convergence of his method. Finally, he discusses a modification of his approach where the choice of the mentioned control vectors does not adversely influence the properties of his method.