The use of the local truncation error to improve arbitrary-order finite elements for the linear wave and heat equations

A New Numerical Approach to the Solution of Partial Differential Equations With Optimal Accuracy on Irregular Domains and Cartesian Meshes

“…However, the improvement in the order of accuracy for the high-order elements in [2,49,50,39,48] is not optimal. In [21,22,19] the order of accuracy of the high-order elements on rectangular domains has been improved to 4p and this order is optimal at a given width of stencil equations.…”

“…A new numerical approach suggested in this paper is the generalization of our previous numerical algorithms developed for the improvement of accuracy of linear and high-order finite element techniques for wave propagation problems and heat transfer problems for regular rectangular domains with uniform meshes. For example, in [23,25,24,19,18,20] we have improved the accuracy of the linear finite elements and the high-order isogeometric elements used for wave propagation from order 2p to order 4p where p is the order of the polynomial basis functions. These techniques have been based of the reduction of the numerical dispersion error.…”

“…Because the numerical dispersion error is based on the existence of exact and numerical harmonic solutions of systems of partial differential equations and its discrete counterpart then the application of these approaches were limited to wave propagation problems for regular rectangular domains with uniform meshes only (the numerical dispersion error is defined on uniform meshes). In [21,22,19] we have improved the accuracy of the linear finite elements and the high-order isogeometric elements for time dependent and time independent heat transfer problems for regular rectangular domains with uniform meshes. In contrast to the use of the numerical dispersion error, the approach in [21,22,19] was based on the minimization of the order of the local truncation error and did not require the existence of exact and numerical harmonic solutions.…”

“…In [21,22,19] we have improved the accuracy of the linear finite elements and the high-order isogeometric elements for time dependent and time independent heat transfer problems for regular rectangular domains with uniform meshes. In contrast to the use of the numerical dispersion error, the approach in [21,22,19] was based on the minimization of the order of the local truncation error and did not require the existence of exact and numerical harmonic solutions. However, the use of uniform meshes in [21,22,19] reduces the application of the technique to rectangular domains and significantly restricts the value of the proposed approach.…”

“…In contrast to the use of the numerical dispersion error, the approach in [21,22,19] was based on the minimization of the order of the local truncation error and did not require the existence of exact and numerical harmonic solutions. However, the use of uniform meshes in [21,22,19] reduces the application of the technique to rectangular domains and significantly restricts the value of the proposed approach. Moreover, in our paper [19] it was shown that the direct application of some techniques with improved accuracy on uniform meshes (e.g., the MIR techniques in [12,51,24]) to nonuniform meshes leads to a significant degradation in the order of accuracy (e.g., by three orders for the wave equation).…”