Implementation and efficiency analysis of an adaptive hp-finite element method for solving boundary value problems for the stationary reaction–diffusion equation

Abstract

An iterative process implementing an adaptive hp-version of the finite element method (FEM) previously proposed by the authors for the approximate solution of boundary value problems for the stationary reaction-diffusion equation is described. The method relies on piecewise polynomial basis functions and makes use of an adaptive strategy for constructing a sequence of finite-dimensional subspaces based on the computation of correction indicators. Singularly perturbed boundary value test problems with smooth and not very smooth solutions are used to analyze the efficiency of the method in the situation when an approximate solution has to be found with high accuracy. The convergence of the approximate solution to the exact one is investigated depending on the value of the small parameter multiplying the highest derivative, on the family of basis functions and the quadrature formulas used, and on the internal parameters of the method. The method is compared with an adaptive h-version of FEM that also relies on correction indicators and with its nonadaptive variant based on the bisection of grid intervals.Keywords: singularly perturbed boundary value problems, stationary one-dimensional reaction-diffusion equations, adaptive methods, correction indicators, hp-version of the finite element method. INTRODUCTIONIn [1] an adaptive method was proposed for finding approximate classical or weak solutions of the Dirichlet boundary value problem for the one-dimensional stationary reaction-diffusion equation formulated as the minimization of a quadratic functional. The goal of adaptation is to construct an approximate solution with a small error uniformly distributed over the integration interval. The method proposed is a variant of the hp-version of the finite element method (FEM) with piecewise polynomial basis functions and an adaptive strategy for constructing a sequence of finite-dimensional subspaces in which the next approximate solution is sought. This strategy is based on the use of correction indicators, i.e., quantities that evaluate the variation occurring in the approximate solution or the minimized functional when the current subspace is expanded or narrowed by adding new basis functions or deleting some old ones. In [1] the general description of the method was given, efficient algorithms for computing correction indicators were designed, and some adaptive strategies were proposed. References to works concerning this subject can be found in [1].The method is intended primarily for problems with solutions having local singularities, specifically, for singularly perturbed boundary value problems. Since the method automatically generates a grid space in which an approximate solution is sought, there is no need to choose a suitable grid or degrees of polynomials ensuring the required accuracy in the case of nonsingular problems as well.In this paper, we describe an iterative process implementing the adaptive method and analyze its efficiency as applied to finding highly accurate approximate solutions to boundary val...