The application of the optimal local truncation error method with unfitted Cartesian meshes to the solution of time-independent partial differential equations for heterogeneous materials

A new numerical technique for the 3-D Poisson equation as well as for the 3-D time-independent elasticity equations for heterogeneous materials with irregular interfaces has been developed. The optimal local truncation error method (OLTEM) with trivial unfitted Cartesian meshes and simple 27-point (similar to the stencils for linear finite elements) and 125-point (similar to the stencils for quadratic finite elements) uniform stencil equations are used. The calculation of the coefficients of the stencil equations is based on the minimization of the local truncation error of the stencil equations and yields the optimal order of accuracy for the new technique. OLTEM with 27-point stencils for the Poisson equation in heterogeneous materials achieves third-order accuracy, which is one order higher compared to linear finite elements. OLTEM with 125-point stencils for the Poisson equation in heterogeneous materials reaches an impressive 11th-order accuracy—an extraordinary eight-order improvement over quadratic finite elements. For the 3-D time-independent elasticity equations in heterogeneous materials with irregular interfaces, OLTEM with 27-point stencils on unfitted Cartesian meshes achieves optimal second-order accuracy. For the first time, a new post-processing procedure has been developed with OLTEM for the calculation of the spatial derivatives of numerical solutions. The spatial derivatives for each grid point are calculated with the help of one compact 27-point stencil or 125-point stencil (the same as for basic computations) for the corresponding grid point and the use of the original PDE. Compared to linear finite elements with similar 27-point stencils, at the engineering accuracy of 0.1% OLTEM decreases the number of degrees of freedom by a factor of greater than 2.9 × 109 for the Poisson equation and by 6.6 × 108 times for the elasticity equations. This substantial reduction significantly decreases computation time. Furthermore, the new approach can be directly applied to other partial differential equations.