Diffusion Coefficients Estimation for Elliptic Partial Differential Equations
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© 2017 Society for Industrial and Applied Mathematics. This paper considers the Dirichlet problem -div(a∇ua) = f on D, ua = 0 on ∂D, for a Lipschitz domain D ⊂ ℝd, where a is a scalar diffusion function. For a fixed f, we discuss under which conditions a is uniquely determined and when a can be stably recovered from the knowledge of ua. A first result is that whenever a ∈ H1(D), with 0 < λ ≤ a ≤ Λ on D, and f ∈ L∞(D) is strictly positive, then ∥a - b∥L2(D) ≤ C∥ua - ub∥H01(D)1/6. More generally, it is shown that the assumption a ∈ H1(D) can be weakened to a ∈ Hs(D), for certain s < 1, at the expense of lowering the exponent 1/6 to a value that depends on s.
author list (cited authors)
Bonito, A., Cohen, A., DeVore, R., Petrova, G., & Welper, G.