Error estimation and adaptivity for finite-volume methods on unstructured triangular meshes: Elliptic heat transfer problems

In this article, a simple and reliable a posteriori error estimate methodology for the finite-volume method on triangular meshes and an adaptive mesh refinement procedure are presented. The proposed error estimate employs a high-order approximation for the scalar at the triangles faces. The estimate technique does not demand expressive computational efforts and memory storage. The adaptive procedure is based on the equal distribution of the error over all the triangles, allowing for suitable local mesh refinements. The error is measured by an H-1 norm, and its convergence behavior is evaluated using four elliptic problems for which the analytical solutions are known. The error differences using analytical and estimate solutions are compared for those problems, and good performance of the adaptive procedure is verified.