Improved Successive Constraint Method Based A Posteriori Error Estimate for Reduced Basis Approximation of 2D Maxwell’s Problem Yanlai Chen1 , Jan S. Hesthaven1 , Yvon Maday2,1 , Jer´onimo Rodr´ıguez3 1 - Division of Applied Mathematics, Brown University, 182 George St, Providence, RI 02912, USA 2 - Universit´ e Pierre et Marie Curie-Paris6, UMR 7598, Laboratoire J.-L. Lions, Paris, F-75005 France 3 - Laboratoire POEMS, 32 Boulevard Victor, 75739 Paris Cedex 15, France
Abstract In a posteriori error analysis of reduced basis approximations to affinely parametrized partial differential equations, the construction of lower bounds for the coercivity and inf-sup stability constants is essential. In [7], the authors presented an efficient method, compatible with an off-line/on-line strategy, where the on-line computation is reduced to minimizing a linear functional under a few linear constraints. These constraints depend on nested sets of parameters obtained iteratively using a greedy algorithm. We improve here this method so that it becomes more efficient due to a nice property, namely, that the computed lower bound is monotonically increasing with respect to the size of the nested sets. This improved evaluation of the inf-sup constant is then used to consider a reduced basis approximation of a parameter dependent electromagnetic cavity problem both for the greedy construction of the elements of the basis and the subsequent validation of the reduced basis approximation. The problem we consider has resonance features for some choices of the parameters that are well captured by the methodology.
1. Introduction In the context of optimization, design or optimal control in many fields including, but not limited to, heat and mass transfer, solid mechanics, acoustics, fluid dynamics and electromagnetics, the numerical simulation of parametric problems written under the weak form: find u(ν) in an Hilbert space X such that a(u(ν), v; ν) = f (v; ν), ∀v ∈ X, (1) Email addresses: Yanlai
[email protected] (Yanlai Chen 1 ),
[email protected] (Jan S. Hesthaven1 ),
[email protected] (Yvon Maday2,1 ),
[email protected] (Jer´ onimo Rodr´ıguez 3 ). Preprint submitted to Elsevier Science
has to be done for many input parameter ν — a P -tuple with moderate P — chosen in a given parameter set D (a closed and bounded subset of RP ). In the previous problem a and f are bilinear and linear forms, respectively, associated to the PDE. The natural hypotheses over a(w, v; ν) that make problem (1) well-posed are – uniform continuity, that is, there exists a uniformly bounded γ(ν) such that |a(w, v; ν)| ≤ γ(ν)kwkX kvkX ,
∀w, v ∈ X, ∀ν ∈ D,
– uniform inf-sup condition 0 < β0 < β(ν) ≡ inf sup
ω∈X v∈X
ka(ω, ·; ν)kX 0 a(ω, v; ν) = inf , ∀ν ∈ D. ω∈X kωkX kvkX kωkX
It follows directly from these assumptions that (1) admits a unique solution. A finite element (FE) discretization of problem (1) is a standard way to approximate its solution uN (ν) ' u(ν) : Given ν ∈ D ⊂ RP , find uN (ν) ∈ X N satisfying aN (uN (ν), v; ν) = f N (v; ν) 1 , ∀v ∈ X N . Here X N is the finite element space approximating X with dim(X N )≡ N . The well-posedness is a consequence of the discrete inf-sup condition, 0 < β0N < β N (ν) ≡ inf
sup
ω∈X N v∈X N
aN (ω, v; ν) , ∀ν ∈ D. kωkX N kvkX N
It is most of the times infeasible to directly solve the finite element problem too many times because of the high marginal cost resulting from the large dimension of the discrete systems to be solved (equal to the dimension N of the finite element space). The reduced basis method (RBM) [2], [5], [11], [13], [14], [16] has emerged as a very efficient and accurate method in this scenario. The fundamental observation that is recognized and exploited by the RBM is the following. Instead of being an arbitrary element of X N , uN (ν) typically resides on Mν = {uN (ν), ν ∈ D} that can be well approximated by a finite-dimensional space whenever the set Mν has a small Kolmogorov width in X. The idea is then to propose an approximation of Mν by W N = span{uN (ν1 ), . . . , uN (νN )}, where, uN (ν1 ), . . . , uN (νN ) are N (
