NEW PROGRESS IN 1D SELF-ADAPTIVE ANALYSIS BASED ON EEP TECHNOLOGY: FROM LINEARITY TO NONLINEARITY

The EEP (Element Energy Projection) method for super-convergence calculation in the post-processing stage of FEM and the self-adaptive strategy based on the EEP technology are fully successful for linear ODEs and hence pave the way for extending to the self-adaptive analysis of nonlinear ODEs. With recent intensive studying, the technology transfer from linearity to nonlinearity in the self-adaptive analysis of 1D problems has successfully been achieved and the present paper gives a brief overview and report about this progress. A new self-adaptive finite element (FE) strategy for nonlinear ODE problems was proposed.  In this method, by means of linearization, the existing linear self-adaptive strategy based on the EEP method is incorporated directly into the solution of nonlinear ODEs to avoid constructing super-convergent formulae and the self-adaptive algorism for each specific and individual nonlinear problem. As a result, a general and unified self-adaptive algorism was proposed. The numerical examples show that the proposed method is highly efficient, stable, general and reliable with the results satisfying the user-preset error tolerance by maximum norm, and hence can serve as the core theory and the algorithm of an advanced and efficient FE solver for nonlinear ODEs.