Residual-based a posteriori error estimation for multipoint flux mixed finite element methods
For a polygonal resp. polyhedral bounded domain in two resp. three dimensions the authors consider a system of first-order differential equations along with first and second kind boundary conditions. These equations model diffusion transfer or Darcy flow and are discretized using the multipoint flux mixed finite elements (on a shape-regular triangulation). This method, following previous work by \textit{R. Ingram} et al. [SIAM J. Numer. Anal. 48, No. 4, 1281--1312 (2010; Zbl 1228.65225)], employs a special quadrature rule using the vertices as knots and leading to the possibility to eliminate the velocities which are approximated using lowest order Brezzi-Douglas-Marini finite elements. Then a cell-centered finite volume pressure approximation remains. The L\(_2\) error indicators used are based on local residuals, jump terms and local quadrature error indicators (which latter do not appear in the work by \textit{C. Carstensen} [Math. Comput. 66, No. 218, 465--476 (1997; Zbl 0864.65068)]), and are shown (using an auxiliary problem the pressure solution of which is locally postprocessed, an approach which fails on quadrilateral resp. hexahedral elements) to be both efficient and reliable. Numerical results are presented for two examples with uniformly resp. adaptively refined meshes (the latter employing Dörfler marking and longest edge refinement) verifying the theoretical results and showing the advantage of the adaptivity when the domain is of L-form or the permeability is subdomain-wise constant.
scientific article; zbMATH DE number 6619584
scientific article