To Infinity in One Step: Equation Adaptivity for Perfectly Matched Layers (Jonathan) & The Direct Computation of Time-Periodic Solutions of PDEs (Puneet)

Jonathan Deakin & Puneet Matharu (The University of Manchester)

ATB Frank Adams 1,

To Infinity in One Step: Equation Adaptivity for Perfectly Matched Layers (Jonathan)

The widely used perfectly matched layer (PML) method employs a complex coordinate transformation to impose non-reflecting boundary conditions in the finite element solution of wave equations.The coordinate transformation is applied in a layer surrounding the region of interest (the bulk) and in its exact form completely absorbs waves leaving the bulk, eliminating spurious reflections from the outer boundary. However, when the problem is discretised, if the solution in the PML region is under-resolved, spurious reflections are created, limiting the achievable accuracy of the computed solution in the bulk. Conversely, if the solution in the PML region is over-resolved then computational effort is wasted. The optimal balance between bulk and PML refinement is problem dependent and difficult to find \textit{a priori}.

To address this problem we propose a PML which is optimal in the sense that the coordinate transformation ensures the solution varies linearly through the PML, making it trivial to discretise. We present an algebraic method for finding this optimal transformation which utilises information about the solution. While this makes the problem non-linear, we show that we can converge to the exact solution by iterating, using information from the previous solution. This iteration is natural if we are already performing spatial mesh adaptation in the bulk. We show that with this optimal PML, the numerical error is completely controlled by the refinement in the bulk.

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