Adaptive $hp$-finite element method for solving boundary value problems for the stationary reaction–diffusion equation

An adaptive $hp$-finite element method with finite-polynomial basis functions is constructed for finding a highly accurate solution of a boundary value problem for the stationary reaction–diffusion equation. Adaptive strategies are proposed for constructing a sequence of finite-dimensional subspaces based on the use of correction indicators, i.e., quantities evaluating the degree to which a chosen characteristic of the approximate solution varies when the subspace is expanded by adding new test basis functions. Efficient algorithms for computing correction indicators are described. The method is intended for problems whose solutions have a local singularity, for example, for singularly perturbed boundary value problems.