Keywords: computational engineering, multigrid, computational efficiency, optimal computational complexity..

This paper discusses the application of multigrid methods to the solution of computational engineering problems arising in a wide range of application areas. The goal of all multigrid solvers is to obtain an accurate solution in a computational time that is proportional to the number of degrees of freedom that are used in the numerical approximation. In order to achieve this a number of components are required, and each of these are outlined in the paper. A number of key algorithms are also presented in order to demonstrate the simplicity of the multigrid approach for both linear and nonlinear model problems. In addition, extensions to problems involving adaptive mesh refinement and the use of algebraic multigrid (AMG) techniques are described.

Having introduced the fundamental concepts of multigrid, the remainder of the paper is divided into two more parts. The first of these provides a short review of the application of the multigrid technique to a range of engineering problems in fluid flow, solid mechanics and electromagnetics. Common themes that arise include the inclusion of the multigrid technique as an inner step within more general solution algorithms, the widespread use of AMG and frequent attempts to employ multigrid in parallel. This latter aspect is particularly challenging due to the inherent difficulties associated with solving coarse grid problems efficiently on multiple processors.

Finally, this paper provides two case studies taken from the author's own research. The first of these illustrates the application of a multigrid with local mesh refinement. This is in the context of the simulation of a phase change problem using second order finite differences in space and a fully implicit, stiff, time integrator. The second study shows an application of multigrid on a parallel computing system. Here the chosen application is the spreading of a thin liquid drop, which is also governed by a time-dependent nonlinear parabolic system of partial differential equations. Again a second order finite difference scheme is used in space and an implicit second order time-stepping rule is applied. The parallelism is achieved through a geometric decomposition of the problem and the use of columns of "ghost nodes" at the edge of each subdomain so that the inter-processor communication is minimized.

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