Linear Computational Cost Implicit Variational Splitting Solver with Non-regular Material Data for Parabolic Problems

We employ a variational splitting for the Crank-Nicolson method and Pennes bioheat equation modeling the heating of the human head as a result of the cellphone antenna radiation. The solution of the system of equations resulting from the 3D discretization of the implicit time integration scheme with the Crank-Nicolson method has $$\mathcal{O}(N^2)$$ complexity using direct solver, resulting in the exact solution. Iterative solvers (e.g., multi-grid solvers) deliver $$\mathcal{O}(Nk)$$ computational cost resulting in an approximate solution. The alternating direction implicit solver delivers $$\mathcal{O}(N)$$ complexity instead; it provides the exact solution (as the direct solver). Still, it requires a regular tensor product structure of the material data. In this paper, we propose a method for generalizing the linear computational cost alternating direction implicit solver using the Crank-Nicolson scheme into non-regular material data.