A two‐grid characteristic finite volume element method for semilinear advection‐dominated diffusion equations

Abstract

Abstract A two‐grid finite volume element method, combined with the modified method of characteristics, is presented and analyzed for semilinear time‐dependent advection‐dominated diffusion equations in two space dimensions. The solution of a nonlinear system on the fine‐grid space (with grid size h ) is reduced to the solution of two small (one linear and one nonlinear) systems on the coarse‐grid space (with grid size H ) and a linear system on the fine‐grid space. An optimal error estimate in H 1 ‐norm is obtained for the two‐grid method. It shows that the two‐grid method achieves asymptotically optimal approximation, as long as the mesh sizes satisfy h = O ( H 2 ). Numerical example is presented to validate the usefulness and efficiency of the method. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013