Two-dimensional cascadic finite element computations of combustion problems Original Research Article

We present an integrated time-space adaptive finite element method for solving systems of two-dimensional nonlinear parabolic systems in complex geometry. The partial differential system is first discretized in time using a singly linearly implicit Runge-Kutta method of order three. Local time errors for the step size control are defined by an embedding strategy. These errors are used to propose a new time step by a PI controller algorithm. A multilevel finite element method with piecewise linear functions on unstructured triangular meshes is subsequently applied for the discretization in space. The local error estimate of the finite element solution steering the adaptive mesh refinement is obtained solving local problems with quadratic trial functions located essentially at the edges of the triangulation. This two-fold adaptivity successfully ensures an a priori prescribed tolerance of the solution. The devised method is applied to laminar gaseous combustion and to solid-solid alloying reactions. We demonstrate that for such demanding applications the employed error estimation and adaption strategies generate an efficient and versatile algorithm.