Two-Grid Discretization Techniques for Linear and Nonlinear PDEs

A number of finite element discretization techniques based on two (or more) subspaces for nonlinear elliptic partial differential equations (PDEs) is presented. Convergence estimates are derived to justify the efficiency of these algorithms. With the new proposed techniques, solving a large class of nonlinear elliptic boundary value problems will not be much more difficult than the solution of one linearized equation. Similar techniques are also used to solve nonsymmetric and/or indefinite linear systems by solving symmetric positive definite (SPD) systems. For the analysis of these two-grid or multigrid methods, optimal Lᵖ error estimates are also obtained for the classic finite element discretizations.