Saddle point least squares for the reaction–diffusion problem

Abstract

We consider a mixed variational formulation for the reaction–diffusion problem based on a saddle point least square approach with an optimal test norm and nonconforming trial spaces. An Uzawa type iterative process for solving the discrete mixed formulations is proposed and choices for discrete stable spaces are provided. The implementation requires a nodal basis only for the test space, and assembly of a global saddle point system is avoided. For the test space, we use piecewise linear spaces of functions on Shishkin type meshes that provide almost optimal approximation in the standard symmetric elliptic formulation. Our saddle point least squares method has the advantage that the order of approximation of the solution in a balanced norm is improved if compared with the standard variational approach. Numerical results are included to support the proposed method.

Keywords

• Least squares
• Saddle point systems
• Mixed methods
• Multilevel methods
• Uzawa algorithm
• Conjugate gradient