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Multigrid third‐order least‐squares solution of Cauchy–Riemann equations on unstructured triangular grids
Author(s) -
Nishikawa H.
Publication year - 2006
Publication title -
international journal for numerical methods in fluids
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.938
H-Index - 112
eISSN - 1097-0363
pISSN - 0271-2091
DOI - 10.1002/fld.1287
Subject(s) - overdetermined system , multigrid method , mathematics , riemann solver , discretization , finite volume method , solver , norm (philosophy) , residual , unstructured grid , triangle mesh , vertex (graph theory) , cauchy distribution , mathematical optimization , mathematical analysis , grid , partial differential equation , geometry , algorithm , graph , discrete mathematics , physics , political science , mechanics , law , polygon mesh
In this paper, a multigrid algorithm is developed for the third‐order accurate solution of Cauchy–Riemann equations discretized in the cell‐vertex finite‐volume fashion: the solution values stored at vertices and the residuals defined on triangular elements. On triangular grids, this results in a highly overdetermined problem, and therefore we consider its solution that minimizes the residuals in the least‐squares norm. The standard second‐order least‐squares scheme is extended to third‐order by adding a high‐order correction term in the residual. The resulting high‐order method is shown to give sufficiently accurate solutions on relatively coarse grids. Combined with a multigrid technique, the method then becomes a highly accurate and efficient solver. We present some results to demonstrate its accuracy and efficiency, including both structured and unstructured triangular grids. Copyright © 2006 John Wiley & Sons, Ltd.