appear (the uj run over an orthonormal basis of automorphic Laplaceeigenforms), so our formula (Theorem 1) is a duality between such integrals and certain geodesic integrals of u. New integral transformations are involved depending on the Laplace-eigenvalue of u. We invert these integral transformations in Section 5, Theorem 2. We develop the formula for finite volume Fuchsian groups, so (as in...

The occurrence of unphysical negative concentrations in solutions to diffusion equations is a well known and severe problem. Especially in three dimensions finite element discretizations give qualitatively very poor results, while finite volume discretizations are much more stable. We investigate the cause of these instabilities and trace them back to constraints on the mesh. It turns out that ...

A general framework for weak residual error estimators applying to various types of boundary value problems in connection with finite element and finite volume approximations is developed. Basic ideas commonly shared by various applications in error estimation and adaptive computation are presented and illustrated. Some numerical results are given to show the effectiveness and efficiency of the...

We study in this paper a posteriori error estimates for H-conforming numerical approximations of diffusion problems with a diffusion coefficient piecewise constant on the mesh cells but arbitrarily discontinuous across the interfaces between the cells. Our estimates give a global upper bound on the error measured either as the energy norm of the difference between the exact and approximate solu...

Abstract This work proposes a finite volume scheme for two-phase Darcy flow in heterogeneous porous media with different rock types. The fully implicit discretization is based on cell-centered, as well face-centered degrees of freedom order to capture accurately the nonlinear transmission conditions at type interfaces. These play major role dynamics. formulated natural physical unknowns, and no...

We show how a stable and accurate hybrid procedure for fluid flow can be constructed. Two separate solvers, one using high order finite difference methods and another using the node-centered unstructured finite volume method are coupled in a truly stable way. The two flow solvers run independently and receive and send information from each other by using a third coupling code. Exact solutions t...

We analyze V –cycle multigrid algorithms for a class of perturbed problems whose perturbation in the bilinear form preserves the convergence properties of themultigrid algorithm of the original problem.As an application, we study the convergence of multigrid algorithms for a covolumemethod or a vertex–centered finite volume element method for variable coefficient elliptic problems on polygonal ...