Papers › Reduced order modeling for elliptic problems with high contrast diffusion coefficients
Reduced order modeling for elliptic problems with high contrast diffusion coefficients
Albert Cohen, Matthieu Dolbeault, Agustin Somacal, Wolfgang Dahmen
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We consider the parametric elliptic PDE -div (a(y)∇u)=f on a spatial domain Ω, with a(y) a scalar piecewise constant diffusion coefficient taking any positive values y=(y₁, …, y_d)∈]0,∞[ᵈ on fixed subdomains Ω₁,…,Ω_d. This problem is not uniformly elliptic as the contrast κ(y)=maxyⱼ/minyⱼ can be arbitrarily high, contrarily to the Uniform Ellipticity Assumption (UEA) that is commonly made on parametric elliptic PDEs. Based on local polynomial approximations in the y variable, we construct local and global reduced model spaces Vₙ of moderate dimension n that approximate uniformly well all solutions u(y). Since the solution u(y) blows as y→0, the solution manifold is not a compact set and does not have finite n-width. Therefore, our results for approximation by such spaces are formulated in terms of relative H¹₀-projection error, that is, after normalization by u(y)_(H¹₀). We prove that this relative error decays exponentially with n, yet exhibiting the curse of dimensionality as the number d of subdomains grows. We also show similar rates for the Galerkin projection despite the fact that high contrast is well-known to deteriorate the multiplicative constant when applying Cea's lemma. We finally establish uniform estimates in relative error for the state estimation and parameter estimation inverse problems, when y is unknown and a limited number of linear measurements ℓᵢ(u) are observed. A key ingredient in our construction and analysis is the study of the convergence of u(y) to limit solutions when some of the parameters yⱼ tend to infinity.
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