Collaborating PDE solvers
type: Item
instance of: scholarly article
title: Collaborating PDE solvers
zbMATH Open document ID: 0767.65086
DOI: 10.1016/0168-9274(92)90045-F
author: H. S. McFaddin
author: John R. Rice
published in: Applied Numerical Mathematics
publication date: 1992-10-25T00:00:00Z
review text: The authors describe a new methodology for solving systems of partial differential equations. They explain the methodology in the case of two- dimensional second-order linear elliptic problems of the form \(L_ i{\mathbf u}_ i = f_ i \text{ for }(x, y) \in D_ i\), \(M_ i{\mathbf u}_ i = g_ i \text{ for }(x, y) \in \partial D_ i\), \(i=1,2,...,m\), where \(L_ i\) is a linear second-order elliptic partial differential operator, \({\mathbf u}_ i, f_ i\), and \(g_ i\) are functions, \(D_ i\) is a subdomain of the entire domain \(D\) where the problem is stated, \(M_ i\) is a linear first-order operator (defining the boundary conditions) and \(\partial D_ i\) denotes the boundary of \(D_ i\). Such a problem may be obtained from a general problem of the form \( L{\mathbf u}=f\) for \((x, y) \in D \) with boundary conditions of the form \(M{\mathbf u} =g\) for \((x, y) \in \partial D\) by dividing the domain \(D\) into many parts \(D_ i\) and formulating the smoothness (or some other ''interface adjusting'') conditions along the interface between each two neighbour subdomains as well as boundary conditions on the related pieces of the boundary of \(D\). The traditional approach to the above general source problem in the case of a very complex shape of the domain \(D\) usually requires first the discretization of the problem (in the space and/or time variable(s) ) and then decompose the obtained algebraic (finite difference) system or/and its discrete domain to simplify the process of solution of the entire discrete problem, but usually using the same method (''solver'') to each subproblem. The methodology proposed in the paper consists in the decomposition of the continuous problem (by subdividing the domain \(D\) into pieces having simple geometric shapes and formulating the interface conditions) and then apply (iteratively) a two-step process: (1) solving separately the obtained subproblems on their subdomains (i.e. applying an individual solver assigned to each subdomain…
Mathematics Subject Classification ID: 65N55
Mathematics Subject Classification ID: 65N30
Mathematics Subject Classification ID: 65Y05
Mathematics Subject Classification ID: 65N38
Mathematics Subject Classification ID: 65N06
Mathematics Subject Classification ID: 35J45
zbMATH DE Number: 69576
zbMATH Keywords: systems of partial differential equations
zbMATH Keywords: linear second-order elliptic operator
zbMATH Keywords: finite difference
zbMATH Keywords: domain decomposition
zbMATH Keywords: second order linear elliptic problem
zbMATH Keywords: domain subdivision
zbMATH Keywords: interface conditions adjusting
zbMATH Keywords: interface relaxation
zbMATH Keywords: discretization after subdivision
zbMATH Keywords: RELAX sofware system
zbMATH Keywords: parallel computation
zbMATH Keywords: complicated domain shape
zbMATH Keywords: enormously complex problems
reviewed by: S.Ząbek
describes a project that uses: RELAX
MaRDI profile type: Publication
cites work: Q4733356
cites work: Geometry Related Convergence Results for Domain Decomposition Algorithms
cites work: An Iterative Procedure with Interface Relaxation for Domain Decomposition Methods
cites work: Q3806726
cites work: Q4028029
cites work: Q3210814
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description: scientific article; zbMATH DE number 69576