Fast algorithms for PDEs in simple and complex geometries

Abstract

We review some fast and accurate numerical techniques for solving elliptic equations in complicated domains developed by the author and his colleagues in recent past. These are based on a combination of fast algorithms for regular domains and various domain embedding techniques. The fast algorithms for regular domains are derived from analysis of integral equation approach for solving elliptic equations in two-and three-dimensions. These algorithms are very accurate and easy to implement on serial as well as parallel computers. For an irregular domain, first the domain is embedded in a regular domain and then the problem is solved in the regular domain using either boundary or distributed control techniques and the fast algorithm for the regular domain. There are more additional considerations in these methods that make the method more efficient and accurate. We do not go into these details in this short paper.One of the key contributing ideas of the author in this broad set-up is the way fast algorithms in regular domains for various elliptic equations are derived. This algorithm was originally conceived during the course of subsonic airfoil design using complex Beltrami equation formulation of subsonic compressible flow equations [1]. We show the basic idea of the algorithm through the following simple model problem in the real plane.where Ω is a unit disk. The solution of this equation can be written asand v(x) is the solution of the following problem:As seen above, computation of u(x) requires computing v(x) and F (x) for x ∈ Ω. The function v(x), x ∈ Ω can be computed very efficiently in complicated domains using boundary element method. How does one compute singular integrals F (x) efficiently and accurately ? Straight-forward computation by quadrature-based method is expensive because of O(N 2 ) per point computational complexity where N is the total number of nodes in the domain. Moreover, the method is not very accurate due to singular nature of the kernel. Our fast and accurate algorithm in circular and annular domains arises from carrying out the computation of the integral in the Fourier domain in combination with careful analysis since the kernel is singular [8]. With M points in the radial direction and N points in the circular direction, the algorithm (for evaluation of singular integral) that results from this analysis involves evaluating F (x) at each of these M × N points using FFT from its radius dependent Fourier coefficients,which, in turn, are obtained from recursive relations in the radial direction involving Fourier coefficients of the source term in these elliptic equations. Computational complexity behind use of recursive relations is less than O(MN log N ) computational complexity of using of FFT a total of 2M times. This operation count is much smaller than O(M 2 N 2 ) which will be required if these integrals were to be evaluated directly at all these MN points using quadrature-based methods. Thus our FFT-Recursive-Relation based algorithm has theoretical computational complexity of ...