GDSW preconditioners for composite Discontinuous Galerkin discretizations of multicompartment reaction-diffusion problems

Abstract

The aim of the present work is to design, analyze theoretically, and test numerically, a generalized Dryja-Smith-Widlund (GDSW) preconditioner for composite Discontinuous Galerkin discretizations of multicompartment parabolic reaction-diffusion equations, where the solution can exhibit natural discontinuities across the domain. We prove that the resulting preconditioned operator for the solution of the discrete system arising at each time step converges with a scalable and quasi-optimal upper bound for the condition number. The GDSW preconditioner is then applied to the EMI (Extracellular - Membrane - Intracellular) reaction-diffusion system, recently proposed to model microscopically the spatiotemporal evolution of cardiac bioelectrical potentials. Numerical tests validate the scalability and quasi-optimality of the EMI-GDSW preconditioner, and investigate its robustness with respect to the time step size as well as jumps in the diffusion coefficients.

Figures (8)

• Figure 1: Overlap between two overlapping subdomains $\Omega'_{j_1}$ and $\Omega'_{j_2}$ for the multicompartment problem described in system (\ref{['eq: multiple parabolic pde']}), 1-dimensional example. On the left, representation of the current minimal overlapping situation. On the right, the considered partition of unity basis function $\chi_{j_1}$ for subdomain $\Omega'_{j_1}$.
• Figure 2: Vertex and edge sharing, 2-dimensional example. On the left, the vertex $\mathcal{V}^l$ is shared by five non-overlapping subdomains, therefore the set $\mathcal{V}^0_l$ contains five indices $\mathcal{V}^0_l = \{ k_1, k_2, k_3, k_4, k_5 \}$. On the right, the vertices $\underline{\mathcal{V}}^l_i$, $\underline{\mathcal{V}}^l_j$ and $\overline{\mathcal{V}}^l_i$, $\overline{\mathcal{V}}^l_j$ represent the same geometric endpoints of the edge $E_{ij} \subset \partial \Omega_i$, but are referred to different subdomains.
• Figure 3: Left: representation of the situation described in system (\ref{['eq: multiple parabolic pde']}), with only one cell $\Omega_1$ (green) immersed in the extracellular liquid (light blue); the external boundary of the extracellular space $\Omega_0$ is divided into $\Gamma_0^D$ (black, dashed) and $\Gamma_0^N$ (black, solid), with boundary conditions given in (\ref{['eq: external boundary conditions']}). Right: representation of the situation described in system (\ref{['eq: multiple parabolic pde']}) considering two neighbouring cells $\Omega_1$ and $\Omega_2$, with common boundary $E_{1,2}$.
• Figure 4: EMI-GDSW Scalability tests on $[0,5]$ ms. Condition number ($k_2$) and linear iterations (it) at final time $t=5$ ms. Fixed time step $\tau=0.05$. Increasing number of cells from $4$ to $1024$, each discretized with $24\times4$ finite elements.
• Figure 5: EMI-GDSW optimality tests on $[0,5]$ ms. Condition number $k_2$ and linear iterations (it) at the final time $t=5$ ms. Fixed time step size $\tau=0.05$ and number of $4\times 4$ cells, each discretized with increasing number of finite elements.

Theorems & Definitions (7)

• Remark 1
• Remark 2
• Lemma 1
• proof
• Theorem 1
• Remark 3
• proof