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Some application driven fast algorithms developed by the author and his collaborators for elliptic partial differential equations are briefly reviewed here. Subsequent use of the ideas behind development of these algorithms for further development of other algorithms some of which are currently in progress is briefly mentioned. Serial and parallel implementation of these algorithms and their applications to some pure and applied problems are also briefly reviewed.
Some application driven fast algorithms developed by the author and his collaborators for elliptic partial differential equations are briefly reviewed here. Subsequent use of the ideas behind development of these algorithms for further development of other algorithms some of which are currently in progress is briefly mentioned. Serial and parallel implementation of these algorithms and their applications to some pure and applied problems are also briefly reviewed.
2011 •
An algorithm for approximating certain classes of elliptic partial differential equations on a rectangle is presented. The algorithm uses high-order 9-point difference approximations to the Helmholtz-type (fourth-order) or Poisson (sixth-order) equations and the fast Fourier transform. Compared to efficient second-order fast direct, methods for smooth problems, the execution time is reduced by a large factor, typically 50 for the Helmholtz-type equations and over 100 for the Poisson problem. Comparisons with two high-order fast direct methods indicate the superiority of the algorithm.
New Trends in Technologies: Devices, Computer, Communication and Industrial Systems
Recent Development of Fast Numerical Solver for Elliptic Problem2010 •
Lecture Notes in Computer Science
A Fully Scalable Parallel Algorithm for Solving Elliptic Partial Differential Equations2007 •
Journal of Computational Physics
A fast semi-direct method for the numerical solution of non-separable elliptic equations in irregular domains1989 •
1976 •
Journal of Computational and Applied Mathematics
The complexity of numerical methods for elliptic partial differential equations1978 •
1999 •
We present a new class of fast direct solvers for elliptic partial differential equations on adaptively refined meshes. These solvers rely on a combination of standard fast solvers for uniform grids and potential theory. Unlike standard iterative approaches, they have a well-defined operation count. They also preserve the order of accuracy of the uniform grid solver, despite the presence of coarse/fine interfaces.
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