Two-scale product approximation for semilinear parabolic problems in mixed methods

We propose and analyze two-scale product approximation for semilinear heat equations in the mixed finite element method. In order to efficiently resolve nonlinear algebraic equations resulting from the mixed method for semilinear parabolic problems, we treat the nonlinear terms using some interpolation operator and exploit a two-scale grid algorithm. With this scheme, the nonlinear problem is reduced to a linear problem on a fine scale mesh without losing overall accuracy of the final system. We derive optimal order L^∞((0, T];L²(Ω))-error estimates for the relevant variables. Numerical results are presented to support the theory developed in this paper.