Local Refinement In Space Research Articles

Block-centred finite difference methods on rectangular composite grids with refinement in space for parabolic equation

ABSTRACTTwo block-centred finite difference schemes with local refinement in space are introduced and analysed to solve parabolic equation. One scheme is simple approximation scheme, in which piecewise constant interpolation approximation is used for slave nodes. The other is more accurate approximation scheme, with piecewise linear interpolation approximation along the interface. Error estimates in the discrete norm are derived of order for simple approximation scheme and of order for more accurate approximation scheme. Numerical experiments using these two schemes show that the convergence rates are in agreement with the theoretical analysis.

A coupling strategy for adaptive local refinement in space and time with a fixed global model in explicit dynamics

In dynamics, domain decomposition methods (DDMs) enable one to use different spatial and temporal discretizations depending on the physical phenomenon being taken into account. Thus, DDMs provide the analyst with key tools for dealing with problems in which phenomena occur on different temporal and spatial scales. This paper focuses on a less intrusive variation of this type of method which enables the global (industrial) mesh to remain unchanged while the local problem is being refined in space and in time where needed. This property is particularly useful in the case of a local problem whose localization evolves rapidly with time, as is the case for delamination. The downside is that the technique is iterative. The method is presented in the context of linear explicit dynamics, but, as with domain decomposition, its extension to other integration schemes and to nonlinear problems should be possible.

Solving dynamic contact problems with local refinement in space and time

Frictional dynamic contact problems with complex geometries are a challenging task – from the computational as well as from the analytical point of view – since they generally involve space and time multi-scale aspects. To be able to reduce the complexity of this kind of contact problem, we employ a non-conforming domain decomposition method in space, consisting of a coarse global mesh not resolving the local structure and an overlapping fine patch for the contact computation. This leads to several benefits: First, we resolve the details of the surface only where it is needed, i.e., in the vicinity of the actual contact zone. Second, the subproblems can be discretized independently of each other which enables us to choose a much finer time scale on the contact zone than on the coarse domain. Here, we propose a set of interface conditions that yield optimal a priori error estimates on the fine-meshed subdomain without any artificial dissipation. Further, we develop an efficient iterative solution scheme for the coupled problem that is robust with respect to jumps in the material parameters. Several complex numerical examples illustrate the performance of the new scheme.

Finite difference schemes for two-dimensional miscible displacement flow in porous media on composite triangular grids

Considering two-dimensional miscible displacement flow in porous media, the local grid refinement method of a coupled system on triangular cell-centered grids with local refinement in space is studied. Based on the balance equation, finite difference schemes of the coupled equations on composite grids are constructed. Studying their stability and convergence properties, the error estimate in the energy norm is obtained. Finally, a numerical example is given.

Finite difference schemes on grids with local refinement in time and space for parabolic problems I. Derivation, stability, and error analysis

Finite difference schemes for parabolic initial value problems on cell-centered grids in space (rectangular for two space dimensions) with regular local refinement in space as in time are derived and their stability and convergence properties are studied. The construction of the finite difference schemes is based on the finite volume approach by approximation of the balance equation. Thus the derived schemes preserve the mass (or the heat).