The element energy projection(EEP) method for computation of superconvergent resulting in a one-dimensional finite element method(FEM) is successfully used to self-adaptive FEM analysis of various linear problems, based on which this paper presents a substantial extension of the whole set of technology to nonlinear problems.The main idea behind the technology transfer from linear analysis to nonlinear analysis is to use Newton's method to linearize nonlinear problems into a series of linear problems so that the EEP formulation and the corresponding adaptive strategy can be directly used without the need for specific super-convergence formulation for nonlinear FEM. As a result, a unified and general self-adaptive algorithm for nonlinear FEM analysis is formed.The proposed algorithm is found to be able to produce satisfactory finite element results with accuracy satisfying the user-preset error tolerances by maximum norm anywhere on the mesh. Taking the nonlinear ordinary differential equation(ODE) of second-order as the model problem, this paper describes the related fundamental idea, the implementation strategy, and the computational algorithm. Representative numerical examples are given to show the efficiency, stability, versatility, and reliability of the proposed approach.
Based on the newly-developed element energy projection(EEP)method with optimal super-convergence order for computation of super-convergent results,an improved self-adaptive strategy for one-dimensional finite element method(FEM)is pro- posed.In the strategy,a posteriori errors are estimated by comparing FEM solutions to EEP super-convergent solutions with optimal order of super-convergence,meshes are refined by using the error-averaging method.Quasi-FEM solutions are used to replace the true FEM solutions in the adaptive process.This strategy has been found to be sim- ple,clear,efficient and reliable.For most problems,only one adaptive step is needed to produce the required FEM solutions which pointwise satisfy the user specified error toler- ances in the max-norm.Taking the elliptical ordinary differential equation of the second order as the model problem,this paper describes the fundamental idea,implementation strategy and computational algorithm and representative numerical examples are given to show the effectiveness and reliability of the proposed approach.
