3,069 research outputs found
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
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
hp-adaptive discontinuous Galerkin solver for elliptic equations in numerical relativity
A considerable amount of attention has been given to discontinuous Galerkin methods for hyperbolic problems in numerical relativity, showing potential advantages of the methods in dealing with hydrodynamical shocks and other discontinuities. This paper investigates discontinuous Galerkin methods for the solution of elliptic problems in numerical relativity. We present a novel hp-adaptive numerical scheme for curvilinear and non-conforming meshes. It uses a multigrid preconditioner with a Chebyshev or Schwarz smoother to create a very scalable discontinuous Galerkin code on generic domains. The code employs compactification to move the outer boundary near spatial infinity. We explore the properties of the code on some test problems, including one mimicking Neutron stars with phase transitions. We also apply it to construct initial data for two or three black holes
Hot new directions for quasi-Monte Carlo research in step with applications
This article provides an overview of some interfaces between the theory of
quasi-Monte Carlo (QMC) methods and applications. We summarize three QMC
theoretical settings: first order QMC methods in the unit cube and in
, and higher order QMC methods in the unit cube. One important
feature is that their error bounds can be independent of the dimension
under appropriate conditions on the function spaces. Another important feature
is that good parameters for these QMC methods can be obtained by fast efficient
algorithms even when is large. We outline three different applications and
explain how they can tap into the different QMC theory. We also discuss three
cost saving strategies that can be combined with QMC in these applications.
Many of these recent QMC theory and methods are developed not in isolation, but
in close connection with applications
Conforming and nonconforming virtual element methods for elliptic problems
We present in a unified framework new conforming and nonconforming Virtual
Element Methods (VEM) for general second order elliptic problems in two and
three dimensions. The differential operator is split into its symmetric and
non-symmetric parts and conditions for stability and accuracy on their discrete
counterparts are established. These conditions are shown to lead to optimal
- and -error estimates, confirmed by numerical experiments on a set
of polygonal meshes. The accuracy of the numerical approximation provided by
the two methods is shown to be comparable
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