Adaptive Methods

The purpose of this work is to apply previously developed adaptive finite element methodologies to the solution of the Helmholtz equation in exterior domains. In doing so, the viability of adaptivity in reducing the cost of computation will be demonstrated. We compute solutions to a model problem of acoustics in two dimensions, using Galerkin and Galerkin Least-Squares (GLS) finite element formulations with the fully coupled (truncated) DtN boundary condition (Givoli and Keller, 1989; Harari and Hughes, 1994). The development of the DtN boundary condition has allowed the exterior problem to be posed on a finite domain, which enables a finite element analysis. The finite element meshes consist of linear triangles. We present a general hp-adaptive strategy, which allows specification of simultaneous refinement in both the element size h and spectral order p. However, we show results for h-refinement only. In addition, we consider only the Helmholtz equation, which governs the frequency domain; adaptive solutions in the time domain are not addressed. Although adaptivity in the time domain is more complex due to the need to track solution features over time, much of the methodology contained herein could be adapted for that purpose.

Traditionally, the Helmholtz equation in exterior domains has been solved using boundary element formulations applied to the boundary integral form of the equation (see, e.g., Burton and Miller, 1971; van den Berg et al., 1991; Kleinman and Roach, 1974; Seybert and Rengarajan, 1987; Cunefare et al, 1989; Kirkup, 1989; Demkowicz et al., 1991; Demkowicz and Oden, 1994). Another approach is the application of infinite elements to the exterior problem (Bettess, 1977; Burnett, 1995). One of the main issues concerning the choice of solution methodologies is the cost of achieving a desired level of accuracy. Boundary element formulations engender considerable savings in that the problem size is much smaller, requiring meshing of only the domain boundary. However, these formulations require significant costs in equation formation, solution of the linear system (since the coefficient matrix is full and not symmetric), and so on. Harari and Hughes (1992a) have compared the costs of boundary element formulations to the costs of finite element formulations using the fully coupled DtN boundary condition, and have found that finite element methods are competitive. Burnett (1995) has shown a significant cost advantage of the infinite element approach compared to boundary element methods.

There are several ways in which the cost competitiveness of finite element techniques can be increased. The first is to change the finite element discretization. For example, Harari and Hughes (1990, 1991, 1992b) developed a GLS method that provides higher accuracy than Galerkin, and maintains stability even on coarse meshes. We will compare GLS and Galerkin solutions in this paper. A second means of decreasing the cost of finite element methods is to use a local DtN boundary condition (Harari and Hughes, 1991; Bayliss and Turkel, 1980; Givoli, 1991; Givoli and Keller, 1990), which leads to savings in both equation formation and solution time, since much smaller bandwidths are obtained in the coefficient matrix. The penalty for such an approach seems to be a reduced accuracy, or the need for C1-continuous shape functions across element interfaces on the artificial boundary. Still other cost-saving measures are possible. Among these are development of iterative solution techniques and parallelization, as well as reducing the cost of mesh generation. The meshing needs of finite element methods can be formidable, particularly in three dimensions. Mesh generation is currently an area of intensive research, and much progress has been made in recent years (see, e.g., Shephard and Georges, 1991; Schroeder and Shephard, 1989, 1990; Peraire et al., 1988; Cavendish et al., 1985; Lohner and Parikh, 1988; Baker, 1989; Blacker and Stephenson, 1991).

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