A posteriori error estimator and h-adaptive finite element method for diffusion-advection-reaction problems

The goal of this paper is to construct h-adaptive finite element method for the linear and nonlinear diffusion-advection-reaction problems. The main result is the element-wise a posteriori error estimator for the piecewise linear approximations of solutions to 2D problems. On each triangle this discontinuous estimator is a quadratic polynomial, which does not require obtaining stream jumps on the mesh sides. We demonstrate the detailed structure of the estimator for the case of piecewise coefficients of the non-linear boundary value problem. Our strategy of adaptivity is based on the control of local relative error distributions and assumes using the bisection method for a local refinement of the triangulations. We illustrate the proposed h-adaptive scheme by some numerical results obtained for the linear and non-linear problems with the exact solutions.