Seminar in Numerical Analysis: Sanna Mönkölä (University of Jyväskylä)

Controllability method for time-harmonic wave equations

A wide range of numerical methods have been used for solving time-harmonic wave equations. Typically, the methods are based on complex-valued formulations leading to large-scale indefinite linear equations. An alternative is to simulate time-dependent equations in time, until the time-harmonic solution is reached. However, this approach suffers from poor convergence, particularly in the case of large wavenumbers and complicated domains. We accelerate the convergence rate by employing a controllability method. The problem is formulated as a least-squares optimization problem, which is solved by the conjugate gradient algorithm. The efficiency of the method relies on smart discretizations. For spatial discretization we use the spectral element method or the discrete exterior calculus, and for time evolution we consider leap-frog style discretization with non-uniform timesteps or higher-order schemes. For constructing spatially isotropic grids for complex geometries, we use non-uniform polygonal structures imitating the close packing in crystal lattices.

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