In the paper we develop the optimal local truncation error method (OLTEM) with the non-diagonal and diagonal mass matrices on unfitted Cartesian meshes for the 3-D time-dependent wave and heat equations for heterogeneous materials with irregular interfaces. 27-point stencils that are similar to those for linear finite elements are used with OLTEM. There are no unknowns for OLTEM on interfaces between different materials; the structure of the global discrete equations is the same for homogeneous and heterogeneous materials. The calculation of the unknown stencil coefficients is based on the minimization of the local truncation error of the stencil equations, includes the use of the interface conditions and the entire PDE in derivations, and yields the optimal fourth order of accuracy of OLTEM with the non-diagonal mass matrix. This is two order higher than for linear finite elements with similar 27-point stencils. OLTEM with the diagonal mass matrix includes the rigorous calculation of the diagonal mass matrix and provides the optimal second order of accuracy. The 3-D numerical results for heterogeneous materials with irregular interfaces show that at the same number of degrees of freedom, OLTEM is even much more accurate than high-order (up to the 7-th order - the highest order in the commercial code COMSOL) finite elements with much wider stencils. Compared to linear finite elements with similar 27-point stencils, at an accuracy of 0.1% OLTEM decreases the number of degrees of freedom by 550–7300 times. This leads to a huge reduction in computation time.

中文翻译：

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用于求解异质材料三维波和热方程的未拟合笛卡尔网格的最优局部截断误差方法


在本文中，我们开发了最佳局部截断误差法 （OLTEM），该方法在未拟合笛卡尔网格上具有非对角线和对角线质量矩阵，用于具有不规则界面的非均质材料的三维瞬态波和热方程。OLTEM 使用类似于线性有限元的 27 点模板。OLTEM 在不同材料之间的界面上没有未知数;对于均质和非均相材料，全局离散方程的结构是相同的。未知模板系数的计算基于模板方程的局部截断误差的最小化，包括在推导中使用界面条件和整个 PDE，并产生具有非对角质量矩阵的 OLTEM 的最佳四阶精度。这比具有类似 27 点模板的线性有限元高两个数量级。具有对角线质量矩阵的 OLTEM 包括对角线质量矩阵的严格计算，并提供最佳的二阶精度。具有不规则界面的异质材料的三维数值结果表明，在相同的自由度数下，OLTEM 甚至比具有更宽模板的高阶（最高 7 阶 - 商业代码 COMSOL 中的最高阶）有限元精确得多。与具有类似 27 点模板的线性有限元相比，在 0.1% 的精度下，OLTEM 将自由度数减少了 550-7300 倍。这大大减少了计算时间。