On a two-dimensional discontinuous Galerkin discretisation with embedded Dirichlet boundary condition

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.The DG-discretisation aims at a high order of accuracy. We discretize with tensor products of cubic polynomials. By construction, such a DG discretisation is fourth order consistent, both in the interior and at the boundaries. By experiments we show fourth order convergence in the presence of a curved Dirichlet boundary. Stability is proved for the one-dimensional Poisson equation with an embedded boundary condition.To illustrate the possibilities of our DG-discretisation, we solve a convection dominated boundary value problem on a regular rectangular grid with a circular embedded boundary condition [7]. We show how accurately the boundary layer with a complex structure can be captured by means of piece-wise cubic polynomials. The example shows that the embedded boundary treatment is effective

Bilinear forms for the recovery-based discontinuous Galerkin method for diffusion

The present paper introduces bilinear forms that are equivalent to the recovery-based discontinuous Galerkin formulation introduced by Van Leer in 2005. The recovery method approximates the solution of the diffusion equation in a discontinuous function space, while inter-element coupling is achieved by a local L2 projection that recovers a smooth continuous function underlying the discontinuous approximation. Here we introduce the concept of a local “recovery polynomial basis” - smooth polynomials that are in the weak sense indistinguishable from the discontinuous basis polynomials - and show it allows us to eliminate the recovery procedure. The recovery method reproduces the symmetric discontinuous Galerkin formulation with additional penalty-like terms depending on the targeted accuracy of the method. We present the unique link between the recovery method and discontinuous Galerkin bilinear forms

Fourier two-level analysis for higher dimensional discontinuous Galerkin discretisation

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. We show that, for a suitably constructed two-dimensional polynomial basis, point-wise block partitioning gives much better results than the classical cell-wise partitioning. Independent of the mesh size, for Poisson's equation, simple MG cycles, with block Gauss Seidel and symmetric block Gauss Seidel smoothing, yield a convergence rate of 0.4 - 0.6 per iteration sweep for both DG-methods studied

A Discontinuous Galerkin Method for Diffusion Based on Recovery

Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/76555/1/AIAA-2007-4083-971.pd

Discontinuous Galerkin discretisation with embedded boundary conditions

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. Within the framework of structured, regular rectangular grids, we treat curved boundaries according to the principles that underlie the discontinuous Galerkin method. Thus, the high-order DG-discretisation is adapted in the cells with embedded boundaries. We give examples of approximation with tensor products of cubic polynomials. As an illustration, we solve a convection dominated boundary value problem on a complex domain. Although, of course, it is impossible to accurately represent a boundary layer with a complex structure by means of a cubic polynomial, the boundary condition treatment appears quite effective in handling such complex situations

Two-level Fourier analysis of a multigrid approach for discontinuous Galerkin discretisation

In this paper we study a multigrid method for the solution of a linear second order elliptic equation, discretized by discontinuous Galerkin (DG) methods, andwe 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

Discontinuous Galerkin discretisation with embedded boundary conditions

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. Within the framework of structured, regular rectangular grids, we treat curved boundaries according to the principles that underlie the discontinuous Galerkin method. Thus, the high-order DG-discretisation is adapted in the cells with embedded boundaries. We give examples of approximation with tensor products of cubic polynomials. As an illustration, we solve a convection dominated boundary value problem on a complex domain. Although, of course, it is impossible to accurately represent a boundary layer with a complex structure by means of a cubic polynomial, the boundary condition treatment appears quite effective in handling such complex situations

Fourier two-level analysis for discontinuous Galerkin discretization with linear elements

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 fordifferent block-relaxation strategies. In addition to an earlier paper where higher-order methods were studied, here we restrict ourselves to methods using piecewise linear approximations. It is well-known that these methods are unstable if no additional interior penalty is applied.As for the higher order methods, we find that point-wise block-relaxationsgive much better results than the classical cell-wise relaxations. Both for the Baumann-Oden and for the symmetric DG method, with a sufficient interior penalty, the block relaxation methods studied (Jacobi, Gauss-Seidel and symmetric Gauss-Seidel) all make excellent smoothing procedures in a classical multigrid setting. Independent of the mesh size, simple MG cycles give convergence factors 0.2 -- 0.4 per iteration sweep for the different discretizations studied