Collaborating PDE solvers (Q1195365)

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, e.g. an individual discretization strategy) and (2) relaxing the interface conditions (i.e. adjusting the function values along the interface so as to better satisfy the interface conditions). One of the main motivations to such an approach is the possibility to apply parallel computers. This motivation to subdivide the original problem may be reasonable also in the case of a simple (e.g. rectangular) shape of the domain \(D\). The authors present the main components of the methodology, particularly discussing how to relax interface conditions. They discuss the application of the approach to more complex partial differential problems and the possibility to apply it to some ''grand challenges'' of computational science, requiring the use of supercomputers, e.g. to the accurate simulation of a complete vehicle or to the simulation of a tank battle. The two above approaches (i.e. ''discretize, then decompose'' and ''decompose first'') are compared and the usefulness of the second one is experimentally (there are no proofs of the effectiveness of the proposed methodology known to the authors) explored. For this purpose the authors developed a prototype computer software system RELAX. It is a system to support high-level interfaces and an object-oriented framework for sets of problem solving modules. These modules are ''encapsulated'' into software objects having their own numerical methods and other tools, e.g. to interact with users; they interact one with another only through the RELAX framework. So each subproblem may be solved in an independent and parallel way.