【分享】Towards a New Implementation of a Nonlinear Solver for FEMLAB

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Partial differential equations form the mathematical foundation for a host of important areas in engineering and physics. FEMLAB R
1 provides a powerful interactive environment for modeling and solving scientific and engineering problems which base on partial differential equations.
Using FEMLAB R
one can model strongly nonlinear coupled multi-physics applications with ease. There is no inherent limitation on the simultaneous simulation of many physical phenomena.
The present version (the latest release 1.1 dates from October 1999) can handle (systems of) second order partial differential equations in one and two space dimensions. The underlying discretization scheme is a finite element method based on piecewise linear triangular elements (and piecewise linear elements in one dimension, respectively).
The FEMLAB R
system is implemented within the MATLAB R
2 environment. The latter is an interactive integrated technical computing environment that combines numerical computation, advanced graphics and visualization, and a high-level programming language. Its open design
makes MATLAB R
easily extensible, and these extensions are platform independent.
With the growing feedback from users of FEMLAB R
, there were repeatedly met highly nonlinear applications (e.g., semiconductor device design, non-Newtonian flows) where the implemented nonlinear solver could not provide a converged solution. Therefore, the need for a
new implementation arose which should be capable of handling highly nonlinear problems. The present report studies different algorithmic approaches. Section 2 provides an overview about different approaches for a robust and efficient realization of the Newton method for solving
nonlinear systems of equations. Section 3 gives some implementational details and design decisions.
Finally, in Section 4, we report about the performance of different algorithms for a test set of highly nonlinear problems whose selection was biased towards the final goal of implementing a solver within the FEMLAB R
environment.