Local Defect Correction

Examples of partial differential equations with solutions that are rapidly varying functions of the spatial or temporal coordinates appear e.g. in combustion, shock hydrodynamics or transport in porous media. For boundary value problems with solutions that have one or a few small regions with high activity, a fine grid is needed in regions with high activity, whereas a coarser grid would suffice in the rest of the domain. Rather than using a truly nonuniform grid, we study a method called local defect correction (LDC) that is based on local uniform grid refinement.

Measurements on a Bunsen flame with cooled walls

Advantages of LDC include the usage of simple data structures and simple accurate discretisation stencils. In the LDC method, the discretisation on the composite grid is based on a combination of standard discretisations on several uniform grids with different grid sizes that cover different parts of the domain.

The LDC method is an iterative process: a basic global discretization is improved by local discretizations defined in subdomains. The update of the coarse grid solution is achieved by adding a defect correction term to the right hand side of the coarse grid problem. At each iteration step, the process yields a discrete approximation of the continuous solution on the composite grid. The discrete problem that is actually being solved is an implicit result of the iterative process. Therefore, the LDC method is both a discretization method and an iterative solution method. LDC can be combined with finite difference discretizations, finite volume discretizations and finite element discretizations.

In a straightforward generalization of the LDC algorithm for finite differences the discrete conservation property, which is one of the main attractive features of a finite volume method, does not hold for the composite grid approximation. We have developed a modified LDC method, which is based on a special form of the defect correction term used in the right hand side of the coarse grid problem. Due to this finite volume adapted defect correction term, the conservation property is preserved in the discretization on the composite grid as is illustrated by the figure below. The composite grid discretization preserves the conservation property at all control volumes.

A local defect correction algorithm implicitly yields a
discretization on a composite grid that consists of a global
coarse grid with one or more regions of local uniform
refinement by applying discretizations on the uniform
grids only

We have extended the standard LDC method to a technique to discretize and solve elliptic boundary value problems on composite grids found by adaptive grid refinement. In this technique, we only use standard discretizations on rectangular tensor-product grids. The full algorithm is obtained by successively adding adaptivity, multi-level refinement, domain decomposition and regridding to the standard LDC algorithm. A procedure for adaptive gridding introduced by Bennett and Smooke is combined with the LDC algorithm. Based on a weight function, that measures the smoothness of the solution of the partial differential equation under consideration, high activity areas are determined and flagged for refinement.

Based on a weight function
boxes are flagged for refinement

The properties above have been illustrated by successfully applying the adaptive multi-level LDC algorithm with domain decomposition to a combustion problem. The mathematical model is a system of nonlinear partial differential equations with strongly nonlinear chemical source terms. The solutions of the system have large gradients in a very small part of the domain and are smooth elsewhere. The simulation results have been compared to those found in literature, and indeed agree well.

Temperature (left side) and CH4 concentration (right)
in a premixed methane/air flame

Our simulation results show that all dependent variables except for the nitrogen mass fraction have large gradients in the flame zone. A remarkable characteristic of the Bunsen flame problem is that the size of the flame increases on the finer grids. The increase is strongest when the first level of refinement is added. The flame length still increases, but less rapidly, with additional refinement. The structure of the flame is similar on all grids.

Maximum level 0 (left side) and 1 (right)

Maximum level 2 (left side) and 3 (right)

Plots of the methane mass fraction
on the finest level for the LDC experiments

LDC is one of the main areas of research for the Scientific Computing Group. For this reason the people working on LDC have regular meetings, the so-called LDC Discussion Clubs. For more information, please visit the LDC Discussion Club page.

For further reference, see the following PhD theses

• M.J.H. Anthonissen
  Local Defect Correction Techniques: Analysis and Application to Combustion
• V. Nefedov
  Numerical Analysis of Viscous Flow using Composite Grids
• P.J.J. Ferket
  Solving Boundary Value Problems on Composite Grids with an Application to Combustion

Contact: dr.ir. M.J.H. Anthonissen