228 research outputs found
An ε -Uniform Numerical Method for a System of Convection-Diffusion Equations with Discontinuous Convection Coefficients and Source Terms
In this paper, a parameter-uniform numerical method is suggested to solve a system of singularly perturbed convection-diffusion equations with discontinuous convection coefficients and source terms subject to the Dirichlet boundary condition. The second derivative of each equation is multiplied by a distinctly small parameter, which leads to an overlap and interacting interior layer. A numerical method based on a piecewise uniform Shishkin mesh is constructed. Numerical results are presented to support the theoretical results
Numerical approximations to a singularly perturbed convection-diffusion problem with a discontinuous initial condition
A singularly perturbed parabolic problem of convection-diffusion type with a discontinuous initial condition is examined. An analytic function is identified which matches the discontinuity in the initial condition and also satisfies the homogenous parabolic differential equation associated with the problem. The difference between this analytical function and the solution of the parabolic problem is approximated numerically, using an upwind finite difference operator combined with an appropriate layer-adapted mesh. The numerical method is shown to be parameter-uniform. Numerical results are presented to illustrate the theoretical error bounds established in the paper. © 2021, The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature
First order least squares method with weakly imposed boundary condition for convection dominated diffusion problems
We present and analyze a first order least squares method for convection
dominated diffusion problems, which provides robust L2 a priori error estimate
for the scalar variable even if the given data f in L2 space. The novel
theoretical approach is to rewrite the method in the framework of discontinuous
Petrov - Galerkin (DPG) method, and then show numerical stability by using a
key equation discovered by J. Gopalakrishnan and W. Qiu [Math. Comp. 83(2014),
pp. 537-552]. This new approach gives an alternative way to do numerical
analysis for least squares methods for a large class of differential equations.
We also show that the condition number of the global matrix is independent of
the diffusion coefficient. A key feature of the method is that there is no
stabilization parameter chosen empirically. In addition, Dirichlet boundary
condition is weakly imposed. Numerical experiments verify our theoretical
results and, in particular, show our way of weakly imposing Dirichlet boundary
condition is essential to the design of least squares methods - numerical
solutions on subdomains away from interior layers or boundary layers have
remarkable accuracy even on coarse meshes, which are unstructured
quasi-uniform
Numerical solution of singularly perturbed 2-D convection-diffusion elliptic interface PDEs with Robin-type boundary conditions
We consider a singularly perturbed two-dimensional convection-diffusion elliptic interface problem with Robin boundary conditions, where the source term is a discontinuous function. The coefficient of the highest-order terms in the differential equation and in the boundary conditions, denoted by ε, is a positive parameter which can be arbitrarily small. Due to the discontinuity in the source term and the presence of the diffusion parameter, the solutions to such problems have, in general, boundary, corner and weak-interior layers. In this work, a numerical approach is carried out using a finite-difference technique defined on an appropriated layer-adapted piecewise uniform Shishkin mesh to provide a good estimate of the error. We show some numerical results which corroborate in practice that these results are sharp
Analysis of optimal error estimates and superconvergence of the discontinuous Galerkin method for convection-diffusion problems in one space dimension
In this paper, we study the convergence and superconvergence properties of the discontinuous Galerkin (DG) method for a linear convection-diffusion problem in one-dimensional setting. We prove that the DG solution and its derivative exhibit optimal O(hp+1) and O(hp) convergence rates in the L 2 -norm, respectively, when p-degree piecewise polynomials with p ≥ 1 are used. We further prove that the p-degree DG solution and its derivative are O(h2p) superconvergent at the downwind and upwind points, respectively. Numerical experiments demonstrate that the theoretical rates are optimal and that the DG method does not produce any oscillation. We observed optimal rates of convergence and superconvergence even in the presence of boundary layers when Shishkin meshes are used
Efficient Resolution of Anisotropic Structures
We highlight some recent new delevelopments concerning the sparse
representation of possibly high-dimensional functions exhibiting strong
anisotropic features and low regularity in isotropic Sobolev or Besov scales.
Specifically, we focus on the solution of transport equations which exhibit
propagation of singularities where, additionally, high-dimensionality enters
when the convection field, and hence the solutions, depend on parameters
varying over some compact set. Important constituents of our approach are
directionally adaptive discretization concepts motivated by compactly supported
shearlet systems, and well-conditioned stable variational formulations that
support trial spaces with anisotropic refinements with arbitrary
directionalities. We prove that they provide tight error-residual relations
which are used to contrive rigorously founded adaptive refinement schemes which
converge in . Moreover, in the context of parameter dependent problems we
discuss two approaches serving different purposes and working under different
regularity assumptions. For frequent query problems, making essential use of
the novel well-conditioned variational formulations, a new Reduced Basis Method
is outlined which exhibits a certain rate-optimal performance for indefinite,
unsymmetric or singularly perturbed problems. For the radiative transfer
problem with scattering a sparse tensor method is presented which mitigates or
even overcomes the curse of dimensionality under suitable (so far still
isotropic) regularity assumptions. Numerical examples for both methods
illustrate the theoretical findings
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