Volume 12, Issue 2

For the solution of incompressible fluid models with inf-sup stable finite element pairs for velocity and pressure, interpolation operators are desirable which preserve the property of discrete zero divergence and enjoy the same local approximation properties as standard interpolation operators. In this work, we show how an anisotropic interpolation operator can be modified preserving the discrete divergence and maintaining certain anisotropic interpolation properties. Beside the construction of such an operator for special anisotropic meshes, we discuss the applicability of anisotropic grid resolution of boundary layers for incompressible low-turbulent flow problems.

A finite difference scheme on special piecewise-uniform grids condensing in the interior layer is constructed for a singularly perturbed parabolic convection-diffusion equation with a discontinuous right-hand side and a multiple degenerating convective term (the convective flux is directed into the domain). When constructing the scheme, monotone grid approximations, similar to those developed and justified earlier by authors for a problem with a simple degenerating convective term, are used. Using the known technique of numerical experiments on embedded meshes, it is numerically verified that the constructed scheme converges ε-uniformly in the maximum norm at the convergence rate close to one.

We analyse the error behaviour of a diffuse-domain approximation of an elliptic differential equation. In one dimension and for a half-plane problem in two dimensions an approximation quality of order one in the interface parameter is shown. Some supporting numerical experiments are also presented.

We apply a combined finite-element finite-volume method on a noncoercive elliptic boundary value problem. The proposed method is based on triangulations of weakly acute type and a secondary circumcentric subdivision. The properties of the continuous problem, that the kernel is one-dimensional and spanned by a positive function, are preserved in the discrete case. A priori error estimates of first order in the H¹-norm are shown for sufficiently small mesh sizes. Numerical test examples confirm the theoretical predictions.

A singularly perturbed convection-diffusion problem is considered. In certain circumstances the solution is shown to be much smaller than its maximum outside a neighborhood of a subcharacteristic curve.

A semilinear second-order singularly perturbed parabolic equation in one space dimension is considered. For this equation, we give computable a posteriori error estimates in the maximum norm for a difference scheme that uses Backward-Euler in time and central differencing in space. Sharp L¹-norm bounds for the Green's function of the parabolic operator and its derivatives are derived that form the basis of the a posteriori error analysis. Numerical results are presented.

A finite difference scheme on special piecewise-uniform grids condensing in tA linear singularly perturbed interior turning point problem with a continuous convection coefficient is examined in this paper. Parameter uniform numerical methods composed of monotone finite difference operators and piecewise-uniform Shishkin meshes, are constructed and analysed for this class of problems. A refined Shishkin mesh is placed around the location of the interior layer and we consider disrupting the centre point of this fine mesh away from the point where the convection coefficient is zero. Numerical results are presented to illustrate the theoretical parameter-uniform error bounds established.

We consider a time-dependent convection diffusion equation in the transport dominated case. As a stabilization method in space we propose a new variant of Local Projection Stabilization (LPS) which uses special enriched bubble functions such that L²-orthogonal local basis functions can be constructed. L²-orthogonal basis functions lead to a diagonal mass matrix which is advantageous for time discretization. We use the discontinuous Galerkin method of polynomial order one for the discretization in time which is superconvergent of order three at the endpoints of the time intervals. In order to avoid the remaining oscillations in the LPS-solution we add for each time step in the space discretization an extra shock capturing term which acts only locally on those mesh cells where an error-indicator is relatively large. The novelty in the shock capturing term is that the scaling factor in front of the additive diffusion term is computed from a low order post-processing error. As a result we obtain both, an oscillation-free discrete solution and the information about the local regions where this solution is still inaccurate due to some smearing. The latter information can be used to create in each time step an adaptively refined space mesh. Whereas the numerical experiments are restricted to one space dimension the proposed ideas work also in the multi-dimensional spatial case. The numerical tests show that the discrete solution with shock capturing is oscillation-free and of optimal accuracy in the regions outside of the shock.