Displaying similar documents to “Some properties of the discontinuous Galerkin method for one-dimensional singularly perturbed problems.”

A Petrov-Galerkin approximation of convection-diffusion and reaction-diffusion problems

Josef Dalík (1991)

Applications of Mathematics

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A general construction of test functions in the Petrov-Galerkin method is described. Using this construction; algorithms for an approximate solution of the Dirichlet problem for the differential equation - ϵ u n + p u ' + q u = f are presented and analyzed theoretically. The positive number ϵ is supposed to be much less than the discretization step and the values of p , q . An algorithm for the corresponding two-dimensional problem is also suggested and results of numerical tests are introduced.

Error estimates for linear finite elements on Bakhvalov-type meshes

Hans-Görg Roos (2006)

Applications of Mathematics

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For convection-diffusion problems with exponential layers, optimal error estimates for linear finite elements on Shishkin-type meshes are known. We present the first optimal convergence result in an energy norm for a Bakhvalov-type mesh.

Opposing flows in a one dimensional convection-diffusion problem

Eugene O’Riordan (2012)

Open Mathematics

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In this paper, we examine a particular class of singularly perturbed convection-diffusion problems with a discontinuous coefficient of the convective term. The presence of a discontinuous convective coefficient generates a solution which mimics flow moving in opposing directions either side of some flow source. A particular transmission condition is imposed to ensure that the differential operator is stable. A piecewise-uniform Shishkin mesh is combined with a monotone finite difference...

The combination technique for a two-dimensional convection-diffusion problem with exponential layers

Sebastian Franz, Fang Liu, Hans-Görg Roos, Martin Stynes, Aihui Zhou (2009)

Applications of Mathematics

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Convection-diffusion problems posed on the unit square and with solutions displaying exponential layers are solved using a sparse grid Galerkin finite element method with Shishkin meshes. Writing N for the maximum number of mesh intervals in each coordinate direction, our “combination” method simply adds or subtracts solutions that have been computed by the Galerkin FEM on N × N , N × N and N × N meshes. It is shown that the combination FEM yields (up to a factor ln N ) the same order of accuracy in...