3,425 research outputs found
Composite Finite Elements for Elliptic Boundary Value Problems with Discontinuous Coefficients
In this paper, we will introduce composite finite elements for solving elliptic boundary value problems with discontinuous coefficients. The focus is on problems where the geometry of the interfaces between the smooth regions of the coefficients is very complicated. On the other hand, efficient numerical methods such as, e.g., multigrid methods, wavelets, extrapolation, are based on a multi-scale discretization of the problem. In standard finite element methods, the grids have to resolve the structure of the discontinuous coefficients. Thus, straightforward coarse scale discretizations of problems with complicated coefficient jumps are not obvious. In this paper, we define composite finite elements for problems with discontinuous coefficients. These finite elements allow the coarsening of finite element spaces independently of the structure of the discontinuous coefficients. Thus, the multigrid method can be applied to solve the linear system on the fine scale. We focus on the construction of the composite finite elements and the efficient, hierarchical realization of the intergrid transfer operators. Finally, we present some numerical results for the multigrid method based on the composite finite elements (CFE-MG
High-order numerical methods for 2D parabolic problems in single and composite domains
In this work, we discuss and compare three methods for the numerical
approximation of constant- and variable-coefficient diffusion equations in both
single and composite domains with possible discontinuity in the solution/flux
at interfaces, considering (i) the Cut Finite Element Method; (ii) the
Difference Potentials Method; and (iii) the summation-by-parts Finite
Difference Method. First we give a brief introduction for each of the three
methods. Next, we propose benchmark problems, and consider numerical tests-with
respect to accuracy and convergence-for linear parabolic problems on a single
domain, and continue with similar tests for linear parabolic problems on a
composite domain (with the interface defined either explicitly or implicitly).
Lastly, a comparative discussion of the methods and numerical results will be
given.Comment: 45 pages, 12 figures, in revision for Journal of Scientific Computin
Hybridized CutFEM for Elliptic Interface Problems
We design and analyze a hybridized cut finite element method for elliptic
interface problems. In this method very general meshes can be coupled over
internal unfitted interfaces, through a skeletal variable, using a Nitsche type
approach. We discuss how optimal error estimates for the method are obtained
using the tools of cut finite element methods and prove a condition number
estimate for the Schur complement. Finally, we present illustrating numerical
examples
Partially Penalized Immersed Finite Element Methods for Elliptic Interface Problems
This article presents new immersed finite element (IFE) methods for solving
the popular second order elliptic interface problems on structured Cartesian
meshes even if the involved interfaces have nontrivial geometries. These IFE
methods contain extra stabilization terms introduced only at interface edges
for penalizing the discontinuity in IFE functions. With the enhanced stability
due to the added penalty, not only these IFE methods can be proven to have the
optimal convergence rate in the H1-norm provided that the exact solution has
sufficient regularity, but also numerical results indicate that their
convergence rates in both the H1-norm and the L2-norm do not deteriorate when
the mesh becomes finer which is a shortcoming of the classic IFE methods in
some situations. Trace inequalities are established for both linear and
bilinear IFE functions that are not only critical for the error analysis of
these new IFE methods, but also are of a great potential to be useful in error
analysis for other IFE methods
Numerical homogenization of elliptic PDEs with similar coefficients
We consider a sequence of elliptic partial differential equations (PDEs) with
different but similar rapidly varying coefficients. Such sequences appear, for
example, in splitting schemes for time-dependent problems (with one coefficient
per time step) and in sample based stochastic integration of outputs from an
elliptic PDE (with one coefficient per sample member). We propose a
parallelizable algorithm based on Petrov-Galerkin localized orthogonal
decomposition (PG-LOD) that adaptively (using computable and theoretically
derived error indicators) recomputes the local corrector problems only where it
improves accuracy. The method is illustrated in detail by an example of a
time-dependent two-pase Darcy flow problem in three dimensions
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