Adaptive Analysis Of1D Nonlinear FEM Based On EEP Super-convergent Method

The numerical solution of ODEs (Ordinary Differential Equations) plays animportant role in the modern mechanics and engineering computing, the numericalsolution of nonlinear ODEs is the central challenge among various difficulties. Thepresent dissertation proposed a new self-adaptive finite element (FE) strategy fornonlinear ODE problems. In this method, the existing linear self-adaptive strategybased on the EEP (Element Energy Projection) method is incorporated directly into thesolution of nonlinear ODEs to avoid constructing super-convergent formula and self-adaptive algorism for each specific and individual nonlinear problem. As a result, ageneral and unified self-adaptive algorism was proposed and the prototype of nonlinearODE solver was formed based on the algorism. The main work of this dissertation is asfollows:1. A fundamental nonlinear iteration strategy of Newton type was proposed basedon the week form of nonlinear ODEs. The concept of “ideal linear problem” wasproposed so that the linear self-adaptive strategy can be introduced directly into thesolution of nonlinear ODEs. Combining the above nonlinear iteration and self-adaptivity techniques, a clear, concise and general fundamental strategy was proposed.2. The fundamental strategy was successfully extended to solving nonlinear C0andC1problems self-adaptively. Mathematical analysis and a number of given numericalexperiments show that the algorism based on the fundamental strategy is able to obtaina final adaptive mesh on which the conventional FEM solutions satisfy the user-specified tolerance point-wise with little accuracy redundancy.3. The fundamental strategy was successfully extended to solving nonlinear first-order ODE systems self-adaptively. The solution of first-order ODE systems hasfundamental significance, because any initial and boundary value problems of highorder ODEs can be equivalently converted to first-order ODE systems. The success ofthe self-adaptive strategy for solving first-order ODE systems broadens the range ofsolving nonlinear problems and forms a unified mode of solving nonlinear ODEproblems.4. Some key issues in nonlinear ODE problems, such as the treatment of nonlinearboundary conditions, the choice of initial solution, and the tracking of solution path and critical points on the solution path, were discussed respectively. A Newton type methodwas proposed to treat nonlinear boundary conditions, further improving the function ofthe nonlinear ODE solver; the continuation method was implemented with some ODEconversion techniques; and the solution of critical points on the solution path wasdirectly solved by solving a converted nonlinear ODE problem.A large number of numerical experiments show that the proposed method in thisdissertation is highly efficient, stable and reliable with the results satisfying the user-preset error tolerance by maximum norm, and hence can serve as the core theory andalgorithm of an advanced and efficient FE solver for nonlinear ODEs.