Alternating directions implicit higher-order finite element method for simulations of time-dependent electromagnetic wave propagation in non-regular biological tissues

TL;DR

This paper introduces an efficient ADI-based finite element method for simulating time-dependent electromagnetic wave propagation in complex biological tissues, leveraging isogeometric analysis for computational efficiency.

Contribution

The paper develops a novel ADI splitting scheme within the IGA framework that maintains linear computational cost even with variable tissue properties.

Findings

Solver verified with manufactured solutions

Effective in simulating electromagnetic waves in the human head

Maintains linear computational cost with variable coefficients

Abstract

We focus on non-stationary Maxwell equations defined on a regular patch of elements as considered in the isogeometric analysis (IGA). 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 formulation resulting in discretization with Kronecker product matrices. We take the tensor product structure of the computational patch of elements from the IGA framework as an advantage, allowing for linear computational cost factorization in every time step. We design our solver to target simulations of electromagnetic waves propagations in non-regular biological tissues. We show that the linear cost of the alternating direction solver is preserved when we arbitrarily vary material data coefficients across the computational domain. We verify the solver using the manufactured solution and the problem of propagation of electromagnetic waves on the human head.