The Mathematical Analysis On Element Energy Projection Method

The finite element method is one of the efficient numerical methods to solve partial different equations. But in general the derivatives of the finite element solution are not continuous across the boundary of elements and have low global accuracy. To improve the accuracy of the derivative of the finite element solution became one of the most important research subject recently.In 2004 for second order two-point boundary value problems Professor Yuan proposed the so called element-energy-projection (EEP) method based on mechanical interpretation. The fundamental idea comes form the strategy in conventional matrix displacement method for skeletal structures and the projection theorem in finite element mathematical theory. The numerical examples show high accuracy of the new method. In 2006, Professor Yuan applied the element energy projection method to the finite element computation of fourth order two point boundary value problems. The numerical examples show that the EEP method also works well for the fourth order problems. But there is no strict mathematical analysis for these very attractive results.In this thesis we give a mathematical analysis on the element energy projection method. Our main results are as follows:1. For self-adjoint second order two point boundary value problem, using projection type interpolation and ultra-approximation results we study the pointwise derivative and displacement recovery formula in detail. We obtain their convergence rate exactly. This result corrected Yuan's earlier conclusion.2. For non-self-adjoint second order two point boundary value problem, we derive formula of derivative of exact solution in nodal point. By virtue of orthogonal correction technique we proved the o(h^2k(k≥1)super convergence for the nodal recovery derivative derived by EEP method. This is the highest order superconvergence result for derivative postprocessing at present.3. For the EEP method of fourth order two point boundary value problem, we extend Yuan's method to more general equations and proved the O(h^2k-2) super convergence for recovery bending moments. This is the highest order superconvergence result for fourth problems up to now. For recovery shear forces we proved the super convergence.