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Subject: finite volume schemes ×

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    Geometric error of finite volume schemes for conservation laws on evolving surfaces
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2013) Giesselmann, Jan; Müller, Thomas
    This paper studies finite volume schemes for scalar hyperbolic conservation laws on evolving hypersurfaces of R3. We compare theoretical schemes assuming knowledge of all geometric quantities to (practical) schemes defined on moving polyhedra approximating the surface. For the former schemes error estimates have already been proven, but the implementation of such schemes is not feasible for complex geometries. The latter schemes, in contrast, only require (easily) computable geometric quantities and are thus more useful for actual computations. We prove that the difference between approximate solutions defined by the respective families of schemes is of the order of the mesh width. In particular, the practical scheme converges to the entropy solution with the same rate as the theoretical one. Numerical experiments show that the proven order of convergence is optimal.
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    Convergence of a finite volume scheme to the eigenvalues of a Schrödinger operator
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2007) Koprucki, Thomas; Eymard, Robert; Fuhrmann, Jürgen
    We consider the approximation of a Schrödinger eigenvalue problem arising from the modeling of semiconductor nanostructures by a finite volume method in a bounded domain $OmegasubsetR^d$. In order to prove its convergence, a framework for finite dimensional approximations to inner products in the Sobolev space $H^1_0(Omega)$ is introduced which allows to apply well known results from spectral approximation theory. This approach is used to obtain convergence results for a classical finite volume scheme for isotropic problems based on two point fluxes, and for a finite volume scheme for anisotropic problems based on the consistent reconstruction of nodal fluxes. In both cases, for two- and three-dimensional domains we are able to prove first order convergence of the eigenvalues if the corresponding eigenfunctions belong to $H^2(Omega)$. The construction of admissible meshes for finite volume schemes using the Delaunay-Voronoï method is discussed. As numerical examples, a number of one-, two- and three-dimensional problems relevant to the modeling of semiconductor nanostructures is presented. In order to obtain analytical eigenvalues for these problems, a matching approach is used. To these eigenvalues, and to recently published highly accurate eigenvalues for the Laplacian in the L-shape domain, the results of the implemented numerical method are compared. In general, for piecewise $H^2$ regular eigenfunctions, second order convergence is observed experimentally.