Adaptive FEM for reaction—diffusion equations Original Research Article

An integrated time—space adaptive finite element method for solving mixed systems of nonlinear parabolic, elliptic, and differential—algebraic equations is presented. The approach is independent of the spatial dimension. For the discretization in time we use singly diagonally linearly implicit Runge—Kutta methods of Rosenbrock type. Local time errors for the step size control are defined by an embedded strategy. A multilevel finite element Galerkin method is subsequently applied for the discretization in space. A posteriori estimates of local spatial discretization errors are obtained solving local problems with higher order approximation. The use of hierarchical basis functions allows simplification of the required computations. Two different strategies to obtain the start grid of the multilevel process are compared. The devised method is applied to a solid—solid combustion problem.