Isogeometric Residual Minimization Method (iGRM) with Direction Splitting for Non-Stationary Maxwell Problems

We focus on non-stationary Maxwell equations defined on a regular patch of elements as considered in the isogeometric analysis. We apply the time-integration scheme following the ideas developed by the finite difference community [M. Hochbruck, T. Jahnke, R. Schnaubelt, Convergence of an ADI splitting for Maxwell's equations, Numerishe Mathematik, 2015] to derive a weak formulations resulting in a discretization with Kronecker product matrices. Going further, we investigate the application of the residual minimization (RM) method for stabilization of the Maxwell equations within the isogeometric analysis setup. The residual minimization method is introduced in every time step of the implicit time integration scheme. We introduce the RM in such a way that we preserve the Kronecker product structure of the matrix. We take the tensor product structure of the computational patch of elements from IGA framework as an advantage, allowing for linear computational cost factorization in every time step, with the automatic stabilization guaranteed by the RM method.