Local defect correction with slanting grids

Abstract

The local defect correction (LDC) method is used to solve a convection‐diffusion‐reaction problem that contains a high‐activity region in a relatively small part of the domain. The improvement of the solution on a coarse grid is obtained by introducing a correction term computed from a local fine‐grid solution. This article studies problems where the high‐activity region is covered with a rectangular fine grid not aligned with the axes of the global domain. This study shows that the resulting method is less expensive than both a uniform refinement and tensor product grid method. © 2003 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 20: 1–17, 2004.

Local Rectangular Refinement in Three Dimensions (LRR3D): Development of a Solution-Adaptive Gridding Technique with Application to Convection-Diffusion Problems

“…Steepness parameter S is set to 22.42, resulting in T changing from 10% to 90% of its maximum over a distance of 0.098 length units, measured normal to the high-activity plane. (This choice of S is motivated by the choice of S in the 2-D studies of [16,18,26,27], which also resulted in T changing from 10% to 90% of its maximum over a distance of 0.098.) Temperature T is approximately equal to 2 in the tetrahedral region between the origin and the highactivity plane, and T % 0 in the rest of the domain.…”

“…The material properties of the fluid are temperature-invariant, with a Peclet number of unity, and a position-dependent heat source (i.e., coming from a chemical reaction, perhaps) is present throughout the domain. This application is a 3-D generalization of a problem used to test the performance of 2-D adaptive gridding methods [26,27], including the LRR2D method [16,18]. The velocity throughout the region is speci-…”

Subgrid Stabilized Defect Correction Methods for the Navier–Stokes Equations

“…Since most practical problems do not have enough regularity, the practical importance was not recognized until the work of Hemker [21,20] and Hemker and Koren [23,22]. One current view of the defect correction method is that it allows for a solution that is nearly nonsingular for ill-conditioned problems through stabilization and correction; for a sample of recent works, see, e.g., Altase and Burrage [1], Axelsson and Nikolova [4], Juncu [28], Graziadei, Mattheij, and Boonkkamp [16], Heinrichs [19,18], Desideri and Hemker [6], Nefedov and Mattheij [34], Shaw and Crumpton [37]. For example, when applied to viscoelastic fluid flow (Lee [32]), the defect correction method proved to be the key algorithmic idea for computing with a Weissenberg number beyond which other algorithms failed.…”

Local defect correction in numerical simulation of surface remelting

“…So-called high order compact difference schemes can be used in an LDC technique too [16]. In [9,17] LDC is studied with different grid types. The method is successfully applied to a combustion problem in [1].…”