586 research outputs found
Introduction to discrete functional analysis techniques for the numerical study of diffusion equations with irregular data
We give an introduction to discrete functional analysis techniques for
stationary and transient diffusion equations. We show how these techniques are
used to establish the convergence of various numerical schemes without assuming
non-physical regularity on the data. For simplicity of exposure, we mostly
consider linear elliptic equations, and we briefly explain how these techniques
can be adapted and extended to non-linear time-dependent meaningful models
(Navier--Stokes equations, flows in porous media, etc.). These convergence
techniques rely on discrete Sobolev norms and the translation to the discrete
setting of functional analysis results
Accurate gradient computations at interfaces using finite element methods
New finite element methods are proposed for elliptic interface problems in
one and two dimensions. The main motivation is not only to get an accurate
solution but also an accurate first order derivative at the interface (from
each side). The key in 1D is to use the idea from \cite{wheeler1974galerkin}.
For 2D interface problems, the idea is to introduce a small tube near the
interface and introduce the gradient as part of unknowns, which is similar to a
mixed finite element method, except only at the interface. Thus the
computational cost is just slightly higher than the standard finite element
method. We present rigorous one dimensional analysis, which show second order
convergence order for both of the solution and the gradient in 1D. For two
dimensional problems, we present numerical results and observe second order
convergence for the solution, and super-convergence for the gradient at the
interface
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