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James H. Bramble
Numerical solutions of partial differential equations
For the past 25 years I have been interested in the development of the theoretical foundation of finite-element methods for the approximation of solutions of elliptic and parabolic partial differential equations. Recently I have concentrated on questions concerning rapid solution of large-scale systems that result from such approximations. Such a question is: Among all the theoretically good approximations to a general class of problems, are there some that can be solved efficiently by taking advantage of modern computer architectures such as parallelism? Answers to questions like this one can bring many problems into the realm of practical feasibility. My current research interest is the design of approximations to solutions to problems in partial differential equations that adequately describe the problem and that can be efficiently solved using modern computing power. In particular, I have been most recently interested in Maxwell's Equations and acoustics, including scattering problems.
- A new approximation technique for div-curl systems (with J. E. Pasciak), Math. Comp. 73 (2004), 1739–1762.
- Analysis of a finite PML approximation for the three dimensional time-harmonic Maxwell and acoustic scattering problems (with J. E. Pasciak), Math. Comp. (to appear).
- The approximation of the Maxwell eigenvalue problem using a least squares method (with T. Kolev and J. E. Pasciak), Math. Comp. (to appear).