844 research outputs found
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
Higher-order finite element methods for elliptic problems with interfaces
We present higher-order piecewise continuous finite element methods for
solving a class of interface problems in two dimensions. The method is based on
correction terms added to the right-hand side in the standard variational
formulation of the problem. We prove optimal error estimates of the methods on
general quasi-uniform and shape regular meshes in maximum norms. In addition,
we apply the method to a Stokes interface problem, adding correction terms for
the velocity and the pressure, obtaining optimal convergence results.Comment: 26 pages, 6 figures. An earlier version of this paper appeared on
November 13, 2014 in
http://www.brown.edu/research/projects/scientific-computing/reports/201
A unified immersed finite element error analysis for one-dimensional interface problems
It has been noted that the traditional scaling argument cannot be directly
applied to the error analysis of immersed finite elements (IFE) because, in
general, the spaces on the reference element associated with the IFE spaces on
different interface elements via the standard affine mapping are not the same.
By analyzing a mapping from the involved Sobolev space to the IFE space, this
article is able to extend the scaling argument framework to the error
estimation for the approximation capability of a class of IFE spaces in one
spatial dimension. As demonstrations of the versatility of this unified error
analysis framework, the manuscript applies the proposed scaling argument to
obtain optimal IFE error estimates for a typical first-order linear hyperbolic
interface problem, a second-order elliptic interface problem, and the
fourth-order Euler-Bernoulli beam interface problem, respectively
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