On high-order conforming finite element methods for ill-posed Helmholtz problems


We consider the numerical approximation of the linear ill-posed problem of unique continuation for the Helmholtz equation. We first review the conditional stability of this problem and then discuss high-order conforming finite element methods, with regularization added on the discrete level using gradient jump penalty and Galerkin least squares. The method is shown to converge in terms of the stability of the problem, quantified by the Hölder exponent, and the polynomial degree of approximation. The analysis also takes into account possibly noisy data. Numerical examples illustrating the theory are presented for a Helmholtz version of the classical Hadamard example for two geometric configurations with different stability properties for the continuation problem, including data perturbations of different amplitudes.


Mihai Nechita
“Tiberiu Popoviciu” Institute of Numerical Analysis, Romanian Academy



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