Large Scale Scientific Computing pp 349-367 | Cite as

# Local Uniform Mesh Refinement on Vector and Parallel Processors

Chapter

## Abstract

The numerical solution of two and three dimensional partial differential equations (PDEs) by various discretizations is an important and common problem which requires significant computing resources. In fact, the computational requirements of these problems are so great that the straightforward methods are too expensive in both time and memory for any computer, existing or planned. We will discuss one class of algorithms for this problem with particular emphasis on their suitability for vector and parallel computers.

## Keywords

Load Balance Parallel Computer Linear Array Coarse Grid Fine Grid## Preview

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## References

- [1]I Babuska and Werner Rheinboldt,
*A Posteriori error analysis of finite element solutions for one-dimensional problems*, SIAM Journal on Numerical Analysis, 18/3 June (1981), pp. 565–589.MathSciNetMATHCrossRefGoogle Scholar - [2]R. Bank. A Multi-Level Iterative Method for Nonlinear Elliptic Equations,
*Elliptic Problem Solvers*, Academic Press, 1981. pp. 1–16.Google Scholar - [3]Marsha J. Berger and Joseph Oliger,
*Adaptive mesh refinement for hyperbolic partial differential equations*, Technical Report Manuscript NA-83–02, Stanford University. March 1983.Google Scholar - [4]Marsha J. Berger and Antony Jameson,
*Automatic adaptive grid refinement for the Euler equations*, Technical Report DOE/ER/03077–202. Courant Mathematics and Computing Laboratory, New York University. October 1983.Google Scholar - [5]M. Bieterman and I. Babuska,
*The finite element method for Parabolic Problems I*, Numerische Mathematik, 40 (1982). pp. 339–371.MathSciNetMATHCrossRefGoogle Scholar - [6]John H. Bolstad,
*An adaptive finite difference method for hyperbolic systems in one space dimension*, Technical Report LBL-13287-rev, Lawrence Berkeley Laboratory, December 1982.Google Scholar - [7]A. Brandt,
*Multi-Level Adaptive Solutions to Boundary Value Problems*, Mathematics of Computation. 31 (1977), pp. 333–390.MathSciNetMATHCrossRefGoogle Scholar - [8]S. Davis and J. Flaherty,
*An Adaptive Finite Element Method for Initial-Boundary Value Problems for Partial Differential Equations*, SIAM Journal on Scientific and Statistical Computing, 3 (1982). pp. 6–27.MathSciNetMATHCrossRefGoogle Scholar - [9]Jack Dongarra and Stanley Eisenstat,
*Squeezing the Most out of an Algorithm in CRAY FORTRAN*, ACM Transactions on Mathematical Software, 10/3 (1984). pp. 219–230.CrossRefGoogle Scholar - [10]H. Dwyer, R. Kee. and B. Sanders,
*Adaptive Grid Method for Problems in Fluid Mechanics and Heat Transfer*, AIAA Journal. 18 (1980), pp. 1205–1212.MATHCrossRefGoogle Scholar - [11]Phillip Ein-Dor,
*Grosch’s law re-revisited: CPU power and the cost of Computation*, Communications of the ACM, 28/2 February (1985), pp. 142–151.CrossRefGoogle Scholar - [12]Dennis Gannon,
*Self Adaptive Methods for Parabolic Partial Differential Equations*, Technical Report UIUCDCS-R-80–1020. Univ. of Illinois. 1980.Google Scholar - [13]William Gropp,
*A test of mesh refinement for 2-d scalar hyperbolic problems*, SIAM Journal on Scientific and Statistical Computing. 1/2 June (1980). pp. 191–197.MATHCrossRefGoogle Scholar - [14]William Gropp,
*Local uniform mesh refinement for elliptic partial differential equations*, Technical Report YALE/DCS/RR-278. Yale University. Department of Computer Science, July 1983.Google Scholar - [15]William Gropp,
*Local uniform mesh refinement on loosely-coupled parallel processors*, Technical Report YALE/DCS/RR-352. Yale University. Department of Computer Science. December 1984.Google Scholar - [16]Alexandru Nicolau and Joseph Fisher, Using an oracle to measure parallelism in single instruction stream programs,
*The 14*^{th}*Annual Microprogramming Workshop*, ACM and IEEE Computer Society. 1981, pp. 171–182.Google Scholar - [17]J. Oliger. Approximate Methods for Atmospheric and Oceanographic Circulation Problems.
*Lecture Notes in Physics, 91*, Springer-Verlag. 1979, pp. 171–184.MathSciNetCrossRefGoogle Scholar

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© Birkhäuser Boston 1987