二阶非自伴两点边值问题Galerkin有限元后处理超收敛解答计算的EEP法

An EEP method for post-computation of super-convergent solutions in one-dimensional Galerkin FEM for second order non-self-adjoint Boundary-Value Problem

将一维Ritz有限元法超收敛计算的EEP(单元能量投影)法推广到二阶非自伴常微分方程两点边值问题Galerkin有限元法的超收敛计算。在对精确单元的研究中,发现与Ritz有限元法不同,只要检验函数采用伴随算子方程的解,无论试函数取何形式,在结点处都可得到精确的解函数值。对近似单元的研究表明,EEP法同样适用于Galerkin有限元法,不仅保留了简便易行、行之有效、效果显著的特点,同时也保留了EEP法的特有优点,如:任一点的导数和解函数的误差与结点值的误差具有相同的收敛阶。

The present paper extends the Element Energy Projection （EEP） method, which is very successful in Ritz FEM, to the super-convergent computation in Galerkin FEM for second order non-self-adjoint BVP（Boundary Value Problem）. In the study of exact elements, it has been shown and proved that, as long as the test functions are constructed using the solution of the adjoint differential equation, the element is bound to produce exact nodal solutions no matter what the trial functions are employed. For approximate elements, it has been found out that the EEP method can well be applied to Galerkin FEM for super-convergent calculation of both solution functions and derivatives at any point on an element in post-processing stage. The proposed method is simple, effective and efficient. A large number of numerical examples consistently show that the accuracy for both solution functions and derivatives so calculated is well comparable to that of the nodal solution values.