90,955 research outputs found
A multiscale method for heterogeneous bulk-surface coupling
In this paper, we construct and analyze a multiscale (finite element) method
for parabolic problems with heterogeneous dynamic boundary conditions. As
origin, we consider a reformulation of the system in order to decouple the
discretization of bulk and surface dynamics. This allows us to combine
multiscale methods on the boundary with standard Lagrangian schemes in the
interior. We prove convergence and quantify explicit rates for low-regularity
solutions, independent of the oscillatory behavior of the heterogeneities. As a
result, coarse discretization parameters, which do not resolve the fine scales,
can be considered. The theoretical findings are justified by a number of
numerical experiments including dynamic boundary conditions with random
diffusion coefficients
Stable cell-centered finite volume discretization for Biot equations
In this paper we discuss a new discretization for the Biot equations. The
discretization treats the coupled system of deformation and flow directly, as
opposed to combining discretizations for the two separate sub-problems. The
coupled discretization has the following key properties, the combination of
which is novel: 1) The variables for the pressure and displacement are
co-located, and are as sparse as possible (e.g. one displacement vector and one
scalar pressure per cell center). 2) With locally computable restrictions on
grid types, the discretization is stable with respect to the limits of
incompressible fluid and small time-steps. 3) No artificial stabilization term
has been introduced. Furthermore, due to the finite volume structure embedded
in the discretization, explicit local expressions for both momentum-balancing
forces as well as mass-conservative fluid fluxes are available.
We prove stability of the proposed method with respect to all relevant
limits. Together with consistency, this proves convergence of the method.
Finally, we give numerical examples verifying both the analysis and convergence
of the method
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FE/BE coupling for an acoustic fluid-structure interaction problem. Residual a posteriori error estimates
This is the author's accepted manuscript. The final published article is available from the link below. Copyright © 2011 John Wiley & Sons, Ltd.In this paper, we developed an a posteriori error analysis of a coupling of finite elements and boundary elements for a fluid–structure interaction problem in two and three dimensions. This problem is governed by the acoustic and the elastodynamic equations in time-harmonic vibration. Our methods combined integral equations for the exterior fluid and FEMs for the elastic structure. It is well-known that because of the reduction of the boundary value problem to boundary integral equations, the solution is not unique in general. However, because of superposition of various potentials, we consider a boundary integral equation that is uniquely solvable and avoids the irregular frequencies of the negative Laplacian operator of the interior domain. In this paper, two stable procedures were considered; one is based on the nonsymmetric formulation and the other is based on a symmetric formulation. For both formulations, we derived reliable residual a posteriori error estimates. From the estimators we computed local error indicators that allowed us to develop an adaptive mesh refinement strategy. For the two-dimensional case we performed an adaptive algorithm on triangles, and for the three-dimensional case we used hanging nodes on hexahedrons. Numerical experiments underline our theoretical results.DFG German Research Foundatio
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