A posteriori error estimates for mixed finite element and finite volume methods for problems coupled through a boundary with nonmatching.

*(English)*Zbl 1303.65093Summary: The primary purpose of this paper is to compare the accuracy and performance of two numerical approaches to solving systems of partial differential equations. These equations are posed on adjoining domains sharing boundary conditions on a common boundary interface in the important case when the meshes used on the two domains are nonmatching across the interface. The first, widely used approach is based on a finite volume method employing ad hoc projections to relate approximations on the two domains across the interface. The second approach uses the mathematically founded mortar mixed finite element method. To quantify the performance, we use a goal-oriented a posteriori error estimate that quantifies various aspects of discretization error to the overall error. While the performance difference may not be a surprise in some cases, we believe that there is a perception in a part of the scientific community concerned with multiphysics systems that if the solution is smooth near the interface, then there is little effect from varying the coupling technique. We find that, on the contrary, the error associated with ad hoc coupling approaches may be large in practical situations. Moreover, we also show that mortar methods can be used with black-box component solvers, thus permitting an efficient and practical implementation also within legacy codes.

##### MSC:

65N15 | Error bounds for boundary value problems involving PDEs |

65N08 | Finite volume methods for boundary value problems involving PDEs |

65N30 | Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs |

65N55 | Multigrid methods; domain decomposition for boundary value problems involving PDEs |

35J25 | Boundary value problems for second-order elliptic equations |