2,051 research outputs found
Stability of explicit one-step methods for P1-finite element approximation of linear diffusion equations on anisotropic meshes
We study the stability of explicit one-step integration schemes for the
linear finite element approximation of linear parabolic equations. The derived
bound on the largest permissible time step is tight for any mesh and any
diffusion matrix within a factor of , where is the spatial
dimension. Both full mass matrix and mass lumping are considered. The bound
reveals that the stability condition is affected by two factors. The first one
depends on the number of mesh elements and corresponds to the classic bound for
the Laplace operator on a uniform mesh. The other factor reflects the effects
of the interplay of the mesh geometry and the diffusion matrix. It is shown
that it is not the mesh geometry itself but the mesh geometry in relation to
the diffusion matrix that is crucial to the stability of explicit methods. When
the mesh is uniform in the metric specified by the inverse of the diffusion
matrix, the stability condition is comparable to the situation with the Laplace
operator on a uniform mesh. Numerical results are presented to verify the
theoretical findings.Comment: Revised WIAS Preprin
Efficient Resolution of Anisotropic Structures
We highlight some recent new delevelopments concerning the sparse
representation of possibly high-dimensional functions exhibiting strong
anisotropic features and low regularity in isotropic Sobolev or Besov scales.
Specifically, we focus on the solution of transport equations which exhibit
propagation of singularities where, additionally, high-dimensionality enters
when the convection field, and hence the solutions, depend on parameters
varying over some compact set. Important constituents of our approach are
directionally adaptive discretization concepts motivated by compactly supported
shearlet systems, and well-conditioned stable variational formulations that
support trial spaces with anisotropic refinements with arbitrary
directionalities. We prove that they provide tight error-residual relations
which are used to contrive rigorously founded adaptive refinement schemes which
converge in . Moreover, in the context of parameter dependent problems we
discuss two approaches serving different purposes and working under different
regularity assumptions. For frequent query problems, making essential use of
the novel well-conditioned variational formulations, a new Reduced Basis Method
is outlined which exhibits a certain rate-optimal performance for indefinite,
unsymmetric or singularly perturbed problems. For the radiative transfer
problem with scattering a sparse tensor method is presented which mitigates or
even overcomes the curse of dimensionality under suitable (so far still
isotropic) regularity assumptions. Numerical examples for both methods
illustrate the theoretical findings
Energy-corrected FEM and explicit time-stepping for parabolic problems
The presence of corners in the computational domain, in general, reduces the
regularity of solutions of parabolic problems and diminishes the convergence
properties of the finite element approximation introducing a so-called
"pollution effect". Standard remedies based on mesh refinement around the
singular corner result in very restrictive stability requirements on the
time-step size when explicit time integration is applied. In this article, we
introduce and analyse the energy-corrected finite element method for parabolic
problems, which works on quasi-uniform meshes, and, based on it, create fast
explicit time discretisation. We illustrate these results with extensive
numerical investigations not only confirming the theoretical results but also
showing the flexibility of the method, which can be applied in the presence of
multiple singular corners and a three-dimensional setting. We also propose a
fast explicit time-stepping scheme based on a piecewise cubic energy-corrected
discretisation in space completed with mass-lumping techniques and numerically
verify its efficiency
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