In this final chapter we discuss some important extensions to the methodology presented in details in the previous chapters to motivate the readers to apply the reduced basis methodology to more complex problems of their own interest. The extensions are related to (i) time-dependent problems, (ii) geometrical parametrization of the computational domain and (iii) non-compliant outputs and primal-dual approximations, and (iv) non-coercive problems. Although the discussion of these extensions are accompanied by numerical tests, we conclude with a final numerical example to offer evidence of the potential of the discussed techniques for complex three-dimensional applications.
Time-Dependent ProblemsConsider the following parabolic model problem: Given a parameter value μ ∈ P ⊂ R P , evaluate the output of interestTo keep the presentation simple we assume that the right hand side f and the output functional are independent of the parameter although this assumption can be relaxed. As in the elliptic case, we assume that the bilinear form a( · , · ; μ) is coercive and continuous (2.4), satisfies the affine assumption (3.11) and is time-invariant. We denote by ( · , · ) L 2 (Ω) the L 2 (Ω) scalar product. In the following, we consider a compliant output as