# Reports of Convection dominated flows

Research within the project: Convection dominated flows

* Research within the* *project: **Convection dominated flows*

*An A Posteriori Adaptive Mesh Technique for Singularly Perturbed Convection-Diffusion Problems with a Moving Interior Layer*

G.I. Shishkin, L.P. Shishkina, P.W. Hemker

2003, MAS-E0326, ISSN 1386-3703

We study numerical approximations for a class of singularly^M perturbed problems of convection-diffusion type with a moving^M interior layer.

Download the pdf-file*On a two-dimensional Discontinuous Galerkin discretisation with embedded Dirichlet boundary condition*

M.H. van Raalte

2003, MAS-R0306, ISSN 1386-3703

In this paper we introduce a discretisation of Discontinuous Galerkin (DG) type for solving 2-D second order elliptic PDEs on a regular rectangular grid, while the boundary value problem has a curved Dirichlet boundary. According to the same principles that underlie DG-methods, we adapt the discretisation in the cell in which the (embedded) Dirichlet boundary cannot follow the gridlines of the orthogonal grid.

Download the pdf-file*Discontinuous Galerkin discretisation with embedded boundary conditions*P.W. Hemker, W. Hoffmann, M.H. van Raalte

2003, MAS-R0301, ISSN 1386-3703

The purpose of this paper is to introduce discretisation methods of discontinuous Galerkin type for solving second order elliptic PDEs on a structured, regular rectangular grid, while the problem is defined on a curved boundary. The methods aim at high-order accuracy and the difficulty arises since the regular grid cannot follow the curved boundary. Starting with the Lagrange multiplier formulation for the boundary conditions, we derive variational forms for the discretisation of 2-D elliptic problems with embedded Dirichlet boundary conditions.

Download the pdf-file*Fourier two-level analysis for higher dimensional discontinuous Galerkin*

P.W. Hemker, M.H. van Raalte

2002, MAS-R0227, ISSN 1386-3703

In this paper we study the convergence of a multigrid method for the solution of a two-dimensional linear second order elliptic equation, discretized by discontinuous Galerkin (DG) methods. For the Baumann-Oden and for the symmetric DG method, we give a detailed analysis of the convergence for cell- and point-wise block-relaxation strategies.

Download the pdf-file*Fourier two-level analysis for discontinuous Galerkin discretization with linear elements*P.W. Hemker, W. Hoffmann, M.H. van Raalte

2002, MAS-R0217, ISSN 1386-3703

In this paper we study the convergence of a multigrid method for the solution of a linear second order elliptic equation, discretized by discontinuous Galerkin (DG) methods, and we give a detailed analysis of the convergence for different block-relaxation strategies.

Download the pdf-file*High-order time-accurate schemes for singularly perturbed parabolic convection-diffusion problems with Robin boundary conditions*

P.W. Hemker, G.I. Shishkin, L.P. Shishkina

2002, MAS-R0207, ISSN 1386-3703

The boundary value problem for a singularly perturbed parabolic PDE with convection is considered on an interval in the case of the singularly perturbed Robin boundary condition; the highest space derivatives in the equation and in the boundary condition are multiplied by the perturbation parameter ε. In contrast to a Dirichlet boundary value problem, for the problem under consideration the errors of well known classical methods, generally speaking, grow without bound as ε <<*N*where^{-1}*N*defines the number of mesh points with respect to*x*. The order of convergence for known ε-uniformly convergent schemes does not exceed 1. In this paper, using a defect correction technique we construct ε-uniformly convergent schemes of high-order time-accuracy.

Download the pdf-file*Two-level Fourier analysis of a multigrid approach for discontinuous Galerkin discretisation*P.W. Hemker, W. Hoffmann, M.H. van Raalte

2002, MAS-R0206, ISSN 1386-3703

In this paper we study a multigrid method for the solution of a linear second order elliptic equation, discretized by discontinuous Galerkin (DG) methods, and we give a detailed analysis of the convergence for different block-relaxation strategies. We find that point-wise block-partitioning gives much better results than the classical cell-wise partitioning. Both for the Baumann-Oden and for the symmetric DG method, with and without interior penalty, the block relaxation methods (Jacobi, Gauss-Seidel and symmetric Gauss-Seidel) give excellent smoothing procedures in a classical multigrid setting. Independent of the mesh size, simple MG cycles give convergence factors 0.075 -- 0.4 per iteration sweep for the different discretisation methods studied.

Download the pdf-file*High-order time-accurate schemes for parabolic singular perturbation problems with convection*

P.W. Hemker, G.I. Shishkin, L.P. Shishkina

2001, MAS-R0101, ISSN 1386-3703

We consider the first boundary value problem for a singularly perturbed parabolic PDE with convection on an interval. For the case of sufficiently smooth data, it is easy to construct a standard finite difference operator and a piecewise uniform mesh, condensing in the boundary layer, which gives an ε-uniformly convergent difference scheme. The order of convergence for such a scheme is exactly one and up to a small logarithmic factor one with respect to the time and space variables, respectively. In this paper we construct high-order time-accurate ε-uniformly convergent schemes by a defect correction technique. The efficiency of the new defect-correction scheme is confirmed by numerical experiments.

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