The use of the local truncation error for the increase in accuracy of the linear finite elements for heat transfer problems

A new approach for the increase in the order of accuracy of the linear finite elements used for the time dependent heat equation and for the time independent Laplace equation has been suggested. It is based on the optimization of the coefficients of the corresponding discrete stencil equation with respect to the local truncation error. By a simple modification of the coefficients of the elemental mass and stiffness matrices, the accuracy of the linear finite elements is improved by two orders for the heat equation and by four orders for the Laplace equation. Despite the significant increase in accuracy, the computational costs of the new technique are the same as those for the conventional linear finite elements on a given mesh. 2-D and 3-D numerical examples are in a good agreement with the theoretical results for the new approach and also show that the new linear finite elements are much more accurate than the conventional linear and quadratic finite elements at the same numbers of degrees of freedom.