Analysis of heat transfer problems using a novel low-order FEM based on gradient weighted operation

In this work, a novel numerical method is formulated to improve the accuracy and efficiency of the finite element method (FEM) when using low-order elements in analysis of heat transfer problems. In this method, the problem domain is discretized using linear triangular or tetrahedral elements. For each independent element, a supporting domain that consists of element itself and its adjacent elements sharing common edges/faces is constructed. Based on the Shepard interpolation, a weighted temperature gradient is then obtained, which will be considered to the generalized Galerkin weak form to form the discretized system equations. In present method, gradient weighted coefficients αi are introduced to reconstruct the temperature gradient field. And the key idea of present method is just the temperature gradient reconstruction through gradient weighted operation. A series of numerical examples are studied to examine the performance of present method in heat transfer analysis. Results show that, with a proper value of αi, present method can achieve much better accuracy, higher convergence rate and is more efficient compared with the standard FEM.