Preconditioned continuous and discontinuous Galerkin spline approximations of time dependent electromagnetic problems in multi-patch domains
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We build upon recent work by some of the authors on high-order spline-based geometric methods for the three-dimensional initial boundary value problem of Maxwell's equations. Previous works were restricted to single-patch geometries. To address the computational challenges associated with inverting mass matrices in the multi-patch setting, we extend results for 0-forms mass and stiffness matrices to 1-forms mass matrices associated with the Hodge star operator. Our approach leverages the tensor-product structure of spline spaces to develop a spectrally equivalent preconditioner for the 1-forms mass matrices. Specifically, we construct an efficient preconditioner for a single-patch domain that remains robust with respect to both mesh size and spline degree. In the multi-patch case, we extend this methodology by integrating the single-patch preconditioner with an Additive Schwarz method, ensuring robustness and scalability with respect to the number of patches. The proposed preconditioner achieves near-optimal computational complexity. Finally, we validate the robustness and computational efficiency of the proposed preconditioners through numerical experiments on realistic problems.