Research And Application Of The Space-time Finite Element Method With Interpolated Coefficient For The Nonlinear Parabolic Equations

Nonlinear parabolic differential equation is an important kind of partial differential equations in mathematical physics discipline , Such as reaction diffusion equations, nonlinear schr(o|¨)dinger equation belong to this type。The exact solutions of the equation is difficult to be calculated, and the problem's application is fairly broad, so use the numerical method to evaluate its approximate solution has very important practical significant.But because of the nonlinear properties, it is cause the solution depends on the initial value, numerical calculation result is also not easily to be obtained. This paper is aimed at this type of nonlinear partial differential equation. According to the nonlinear equations properties, put the interpolation coefficient thought into space and time finite element method, compare with the classical nonlinear finite element method, it is a kind of efficient and economical method. That is both in time and space directions, selecting suitable finite element discrimination that continuous in space and in time, then use interpolation polynomials to process the nonlinear term of equations, thus reduce calculation storage and save computing time. Then on the complex space discussed the nonlinear schrodinger partial differential equation.The mainly results of this paper including the following three aspects:First: using the theory of the Sobolev space, the existence and uniqueness of this kind of equation's weak solution is proved. It is illustrating that use this numerical method to solve this kind of partial differential equation is feasible.Second: using Unit orthogonal approximation techniques proved the nonlinear parabolic differential equation basic error estimates in L∞(L2)norm.Third: Applying the above results, discussed the well-posedness problem of the nonlinear Schrodinger equation , achieved the corresponding theoretical results.Finally: give a concrete numerical example to illustrate the correctness of the above theory analysis.