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Published in:
2014 | OriginalPaper | Chapter
Fundamentals of reduced basis method for problems governed by parametrized PDEs and applications
Author : Gianluigi Rozza
Published in: Separated Representations and PGD-Based Model Reduction
Publisher: Springer Vienna
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In this chapter we consider Reduced Basis (
RB
) approximations of parametrized Partial Differential Equations (
PDE
s). The the idea behind
RB
is to decouple the generation and projection stages (Offline/Online computational procedures) of the approximation process in order to solve parametrized
PDE
s in a fast, inexpensive and reliable way. The
RB
method, especially applied to 3D problems, allows great computational savings with respect to the classical Galerkin Finite Element (FE) Method. The standard
FE
method is typically ill suited to (i) iterative contexts like optimization, sensitivity analysis and many-queries in general, and (ii) real time evaluation. We consider for simplicity coercive
PDE
s. We discuss all the steps to set up a
RB
approximation, either from an analytical and a numerical point of view. Then we present an application of the
RB
method to a steady thermal conductivity problem in heat transfer with emphasis on geometrical and physical parameters.