A Flux Recovery Method Taking Integral Form For Evolution Reaction Diffusion Equations

Numerical simulation of fluid flow usually needs us to solve two unknown functions at the same time, one is a scalar function, and the other is a vector function. Mixed finite element methods are often used to deal with such problems. Theory and application of mixed finite element methods have been studied systematically and great successes have been scored. Their outstanding strong point is possessing local conservation. But their shortcomings are that one has to face solving an indefinite symmetric system resulting from the saddle point formulation, and they need lots of computing work. In this thesis, taking one dimension evolution reaction diffusion equation as a model, we systematically study the flux recovery methods taking integral form for evolution problems. The semi-discrete and total discrete schemes will be provided respectively. We always use finite element methods to approximate the space variable in both of schemes. The time variable keeps continuous in the semi-discrete scheme, but we use finite difference method to approximate the time variable in the total discrete scheme. According to the schemes in this paper, the approximate pressure p hand the approximate flux u h are calculated by two separate steps. Firstly the approximate pressure p h is got by Galerkin finite element method; then the approximate flux u h to the exact flux u is obtained by a simple but physically intuitive formula over each finite element. These methods not only effectively overcome the shortcomings of mixed finite element methods and save lots of computing work, but also maintain local conservation property at the element level. The convergences of all schemes have been analyzed strictly. Numerical results show advantages of the methods in this paper.