We propose a matrix-free solver for the numerical solution of the cardiac electrophysiology model consisting of the monodomain nonlinear reaction-diffusion equation coupled with a system of ordinary differential equations for the ionic species. Our numerical approximation is based on the high-order Spectral Element Method (SEM) to achieve accurate numerical discretization while employing a much smaller number of Degrees of Freedom than first-order Finite Elements. We combine vectorization with sum-factorization, thus allowing for a very efficient use of high-order polynomials in a high performance computing framework. We validate the effectiveness of our matrix-free solver in a variety of applications and perform different electrophysiological simulations ranging from a simple slab of cardiac tissue to a realistic four-chamber heart geometry. We compare SEM to SEM with Numerical Integration (SEM-NI), showing that they provide comparable results in terms of accuracy and efficiency. In both cases, increasing the local polynomial degree $p$ leads to better numerical results and smaller computational times than reducing the mesh size $h$. We also implement a matrix-free Geometric Multigrid preconditioner that results in a comparable number of linear solver iterations with respect to a state-of-the-art matrix-based Algebraic Multigrid preconditioner. As a matter of fact, the matrix-free solver proposed here yields up to 45$\times$ speed-up with respect to a conventional matrix-based solver.

翻译：本文提出了一种无矩阵求解器，用于数值求解心脏电生理模型中的单域非线性反应扩散方程，该方程与描述离子种类的常微分方程组相耦合。我们采用高阶谱元法进行数值逼近，从而在使用的自由度数量远少于一阶有限元的情况下，实现高精度的数值离散。通过将向量化与求和因子化相结合，我们得以在高性能计算框架中高效地使用高阶多项式。我们在多种应用场景中验证了该无矩阵求解器的有效性，并进行了从简单心肌组织切片到真实四腔心脏几何模型的一系列电生理模拟。我们将谱元法与带有数值积分的谱元法进行了对比，结果表明二者在精度和效率上具有可比性。在两种方法中，增加局部多项式次数$p$相较于细化网格尺寸$h$，可带来更好的数值结果并缩短计算时间。我们还实现了一种无矩阵几何多重网格预处理器，其线性求解器迭代次数与基于矩阵的代数多重网格预处理器相当。事实上，本文提出的无矩阵求解器相较于传统基于矩阵的求解器，可实现高达45倍的加速。