A Weak Finite Element Method For Singularly Perturbed Two-point Boundary Value Problems In One Dimensional Space

In the field of science and engineering,many practical problems are attributed to the determining solution problem of partial differential equations.However,most partial differential equations are difficult to solve and can only be solved by numerical methods.The numerical method mainly includes finite difference method and finite element method.In recent years,the finite element method has been proposed based on the finite element method.The main idea of the weak finite element method is to replace the classical derivatives of the variational equation with the weak derivatives defined on the space of discontinuous functions,and calculate the variational equation on the weak finite element space.The weak finite element method provides a new idea for the numerical solution of partial differential equation,which is the improvement and development of the existing discontinuous Galerkin finite element method.This paper mainly studies a weak finite element method for singularly perturbed two-point boundary value problems in one dimensional space.We define the weak derivatives and discrete derivatives of discontinuous function on one dimensional area,and then construct the weak finite element space Sh and use it to give the weak finite element approximation format of the singularly perturbation problem.We prove the unique existence and stability of the weak finite element approximate solution,and derive the optimal error estimates in the discrete the H1-norm,the L2-norm and the L?-norm,respectively.Moreover,some superconvergence results are also given.Our numerical experiments show the effectiveness of the weak finite element method for the singularly perturbed two-point boundary value problem.The finite element method in this paper retains the relevant characteristics of the traditional Galerkin finite element method,but also has the new characteristics.For example,the weak finite element space Sh satisfies the weak embedded inequalities;In addition,the discrete weak finite element system can be solved by locally,element by element.These features are not available for the finite element method in multidimensional.The key characteristics of the weak finite element method is allowed to use totally discontinuous finite element function.And the trace of finite element function on element boundaries is independent to the inner value.and it makes this method possess higher flexibility.