34,790 research outputs found
Finite element exterior calculus for parabolic problems
In this paper, we consider the extension of the finite element exterior
calculus from elliptic problems, in which the Hodge Laplacian is an appropriate
model problem, to parabolic problems, for which we take the Hodge heat equation
as our model problem. The numerical method we study is a Galerkin method based
on a mixed variational formulation and using as subspaces the same spaces of
finite element differential forms which are used for elliptic problems. We
analyze both the semidiscrete and a fully-discrete numerical scheme.Comment: 17 page
A modified semi--implict Euler-Maruyama Scheme for finite element discretization of SPDEs with additive noise
We consider the numerical approximation of a general second order
semi--linear parabolic stochastic partial differential equation (SPDE) driven
by additive space-time noise. We introduce a new modified scheme using a linear
functional of the noise with a semi--implicit Euler--Maruyama method in time
and in space we analyse a finite element method (although extension to finite
differences or finite volumes would be possible). We prove convergence in the
root mean square norm for a diffusion reaction equation and diffusion
advection reaction equation. We present numerical results for a linear reaction
diffusion equation in two dimensions as well as a nonlinear example of
two-dimensional stochastic advection diffusion reaction equation. We see from
both the analysis and numerics that the proposed scheme has better convergence
properties than the standard semi--implicit Euler--Maruyama method
Convergence Rate Estimates for the Low Mach and Alfv\'en Number Three-Scale Singular Limit of Compressible Ideal Magnetohydrodynamics
Convergence rate estimates are obtained for singular limits of the
compressible ideal magnetohydrodynamics equations, in which the Mach and
Alfv\'en numbers tend to zero at different rates. The proofs use a detailed
analysis of exact and approximate fast, intermediate, and slow modes together
with improved estimates for the solutions and their time derivatives, and the
time-integration method. When the small parameters are related by a power law
the convergence rates are positive powers of the Mach number, with the power
varying depending on the component and the norm. Exceptionally, the convergence
rate for two components involve the ratio of the two parameters, and that rate
is proven to be sharp via corrector terms. Moreover, the convergence rates for
the case of a power-law relation between the small parameters tend to the
two-scale convergence rate as the power tends to one. These results demonstrate
that the issue of convergence rates for three-scale singular limits, which was
not addressed in the authors' previous paper, is much more complicated than for
the classical two-scale singular limits
Error Analysis of Semidiscrete Finite Element Methods for Inhomogeneous Time-Fractional Diffusion
We consider the initial boundary value problem for the inhomogeneous
time-fractional diffusion equation with a homogeneous Dirichlet boundary
condition and a nonsmooth right hand side data in a bounded convex polyhedral
domain. We analyze two semidiscrete schemes based on the standard Galerkin and
lumped mass finite element methods. Almost optimal error estimates are obtained
for right hand side data , , for both semidiscrete schemes. For lumped mass method, the optimal
-norm error estimate requires symmetric meshes. Finally, numerical
experiments for one- and two-dimensional examples are presented to verify our
theoretical results.Comment: 21 pages, 4 figure
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