Optimal Local Truncation Error Method for Solution of Partial Differential Equations on Irregular Domains and Interfaces Using Unfitted Cartesian Meshes: Review

The review of the optimal local truncation error method (OLTEM) for the numerical solution of PDEs is presented along with some new developments of OLTEM. First, we explain the basic ideas of OLTEM for the 1-D wave equation and then we extend them to the time-dependent PDEs (the scalar wave and heat equations as well as a system of the elastodynamics equations) and to the time-independent PDEs (the Poisson and Helmholtz equations as well as a system of the elastostatics equations) in the 2-D and 3-D cases for homogeneous, inhomogeneous and heterogeneous materials. The main advantages of OLTEM are the optimal (maximum possible) accuracy of discrete equations and the use of unfitted Cartesian meshes for irregular domains and interfaces. For example, for heterogeneous materials with irregular interfaces, OLTEM with 2-D 25-point stencils (similar to those for quadratic finite elements) provides the 11-th and 10-th orders of accuracy for the Poisson and elasticity equations, i.e, a huge increase in accuracy by 8 and 7 orders compared to quadratic finite elements without additional computational costs. Another advantage of OLTEM is a special procedure for the imposition of the boundary and interface conditions without the introduction of additional unknowns. These conditions at a small number of the selected boundary and interface points are added to the local truncation error as the constraints with Lagrange multipliers. This special procedure does not introduce additional unknowns on the boundaries and interfaces (only the unknowns at internal Cartesian grid points are used), does not change the width of cut stencils, allows unfitted meshes and provides a high accuracy of cut stencils. For time-dependent PDEs, OLTEM offers a rigorous approach for the calculation of the diagonal mass matrix in terms of the coefficients of the stiffness matrix that is based on the accuracy considerations. A new OLTEM post-processing procedure for the calculation of the spatial derivatives of the primary function that is based on the use of original PDEs significantly increases the accuracy of the spatial derivatives. For example, we have obtained the 10-th order of accuracy for stresses calculated by OLTEM with 25-point stencils applied to 2-D elastostatics problems with heterogeneous materials and irregular interfaces. New developments of OLTEM related to numerical high-order boundary conditions for cut stencils as well as to the accurate calculation of the primary functions and their derivatives at any point of the domain are presented. The comparison of accuracy of OLTEM and FEM at similar stencils is also analyzed. Numerical results show that at the engineering accuracy, OLTEM can reduce the number of degrees of freedom by 1000-1,000,000 times compared to that for finite elements at similar stencils.