A high-precision scheme for field variables in finite element method

Finite element method (FEM) is the most widely used numerical analysis method, which is an effective solution to all kinds of engineering and scientific problems. However, the interpolation based on shape function often does not have desired accuracy in cases where the internal field variables of the fmite element need to be solved. To deal with this kind of problems, a high-precision solution is proposed, and the basic idea of the solution is the combination of finite element and Taylor expansion. The value of internal field is obtained by establishing and solving linear equations of unknown field variables in the element with respect to known field values at element nodes. The coefficients of each equation depend on the relative position of the points. The results illustrate that the proposed method, which implement a relatively simple algorithm, is applicable to multi-dimension and multi-order elements. It also shows an excellent performance in the application of nonlinear and high-order field functions. This new solution algorithm with high computational efficiency and high precision is expected to serve the intermediate steps of nonlinear computation and satisfy the requirements of high precision post-processing. The method is not limited to FEM and can be extended to all kinds of discrete numerical methods including meshless method.