Application of the over-set grid technique to a model singular perturbation problem

The numerical solution of a singularly perturbed problem, in the form of a two-dimensional convection-diffusion equation, is studied by using the technique of over-set grids. For this purpose the Overture software library is used. The selection of component grids is made on basis of asymptotic analysis. The behavior of the solution is studied for a range of small diffusion parameters. Also the possibilities of rotating the grid with the convection direction is considered. In order to fit global properties of the solution, the composite grid used is made parameter dependent. In view of possible $eps$-uniform convergence, in the resulting composite grid the number of grid points is kept constant for the different values of the small parameter. Only the grid spacing is adapted, depending on the parameters. We see that, even with careful adaptation of the grid, no $eps$-uniform convergence is achieved

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

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

Microcirculatory alterations in ischemia–reperfusion injury and sepsis: effects of activated protein C and thrombin inhibition

Experimental studies in ischemia–reperfusion and sepsis indicate that activated protein C (APC) has direct anti-inflammatory effects at a cellular level. In vivo, however, the mechanisms of action have not been characterized thus far. Intravital multifluorescence microscopy represents an elegant way of studying the effect of APC on endotoxin-induced leukocyte–endothelial-cell interaction and nutritive capillary perfusion failure. These studies have clarified that APC effectively reduces leukocyte rolling and leukocyte firm adhesion in systemic endotoxemia. Protection from leukocytic inflammation is probably mediated by a modulation of adhesion molecule expression on the surface of leukocytes and endothelial cells. Of interest, the action of APC and antithrombin in endotoxin-induced leukocyte–endothelial-cell interaction differs in that APC inhibits both rolling and subsequent firm adhesion, whereas antithrombin exclusively reduces the firm adhesion step. The biological significance of this differential regulation of inflammation remains unclear, since both proteins are capable of reducing sepsis-induced capillary perfusion failure. To elucidate whether the action of APC and antithrombin is mediated by inhibition of thrombin, the specific thrombin inhibitor hirudin has been examined in a sepsis microcirculation model. Strikingly, hirudin was not capable of protecting from sepsis-induced microcirculatory dysfunction, but induced a further increase of leukocyte–endothelial-cell interactions and aggravated capillary perfusion failure when compared with nontreated controls. Thus, the action of APC on the microcirculatory level in systemic endotoxemia is unlikely to be caused by a thrombin inhibition-associated anticoagulatory action