Generalized Green's Functions and the Effective Domain of Influence
An important characteristic of many elliptic problems is that the strength of the effect of a localized
perturbation on the solution decays with the distance to the perturbation. The decay of influence has
important consequences for adaptive mesh refinement because it implies that discretization error in a
particular part of the domain can be reduced by using local mesh refinement near that part. We explain how
these observations suggest a technique for decomposing the solution of an elliptic problem into a set of component
problems in such a way that the maximum number of elements needed in a finite element solution to achieve a desired
accuracy can be decreased significantly. Using a generalized notion of a Green's function, we can quantify the decay of
influence associated to each component problem. This enables each component problem to be solved using a locally refined
mesh with significantly fewer elements that would be required for the full problem. Yet, the solutions of the component
problems can be combined to yield an accurate solution of the original problem. After describing the decomposition
technique, we use a set of examples to demonstrate that significant computational gains are possible when the goal
is to compute multiple quantities of interest and/or to compute quantities of interest that involve globally-supported
information of the solution like average values and norms.
Graduate students: during the seminar I will suggest an excellent research problem for a Ph.D. thesisDonald Estep
Co-Director
IGERT Program for Interdisciplinary Mathematics, Statistics, and
Ecology
http://www.primes.colostate.edu
Professor
Department of Mathematics
http://www.math.colostate.edu/~estep
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