35,118 research outputs found
A Comparison Study of Two Methods for Elliptic Boundary Value Problems
In this paper, we perform a comparison study of two methods (the embedded
boundary method and several versions of the mixed finite element method) to
solve an elliptic boundary value problem
Enforcing the non-negativity constraint and maximum principles for diffusion with decay on general computational grids
In this paper, we consider anisotropic diffusion with decay, and the
diffusivity coefficient to be a second-order symmetric and positive definite
tensor. It is well-known that this particular equation is a second-order
elliptic equation, and satisfies a maximum principle under certain regularity
assumptions. However, the finite element implementation of the classical
Galerkin formulation for both anisotropic and isotropic diffusion with decay
does not respect the maximum principle.
We first show that the numerical accuracy of the classical Galerkin
formulation deteriorates dramatically with increase in the decay coefficient
for isotropic medium and violates the discrete maximum principle. However, in
the case of isotropic medium, the extent of violation decreases with mesh
refinement. We then show that, in the case of anisotropic medium, the classical
Galerkin formulation for anisotropic diffusion with decay violates the discrete
maximum principle even at lower values of decay coefficient and does not vanish
with mesh refinement. We then present a methodology for enforcing maximum
principles under the classical Galerkin formulation for anisotropic diffusion
with decay on general computational grids using optimization techniques.
Representative numerical results (which take into account anisotropy and
heterogeneity) are presented to illustrate the performance of the proposed
formulation
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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