Uniform convergence and preconditioning methods for projection nonconforming and mixed methods for nonselfadjoint and indefinite problems Original Research Article

The purpose of this paper is to prove the existence, uniqueness and uniform convergence of the solutions of so-called projection nonconforming and mixed element methods and the equivalence between projection nonconforming element method and mixed element method with nonquasi-uniform partition for nonselfadjoint and indefinite second order elliptic problems under minimal regularity assumption. Meanwhile, the optimal error estimate of the solution of mixed element method is obtained under H2 smoothness hypothesis for nonselfadjoint and indefinite elliptic problems without H2 regularity. Finally, the discrete compactness result for nonconforming element space with nonquasi-uniform partition is proven, and some preconditioning methods for projection nonconforming and mixed element methods with nonquasi-uniform partition are given. It is proven that the H1-condition number of preconditioned operator is uniformly bounded and its singular values cluster in a relatively small finite interval.