Finite Volume Element Methods On The Shishkin Mesh For The Singularly Perturbed Problems

In this article,the finite volume element methods and their convergences are sys-tematically studied on the Shishkin mesh for two-dimensional singularly perturbed prob-lems.Specifically,the problems include:the singularly perturbed reaction-diffusion problems,the singularly perturbed convection-diffusion problems and fourth-order sin-gularly perturbed elliptic problems.For the singularly perturbed reaction-diffusion problems,we study the bilinear el-ement finite volume method on the Shishkin mesh,and develop theoretical analysis of the stability and convergence of our method.In the convergence analysis,we first intro-duce the discrete H~1semi-norm containing the aspect ratio of the element,and prove its equivalence with the continuous H~1norm.By using the equivalence of H~1norm and the decomposition theorem of the solution,the interpolation weak estimation of the bilinear form is proved.Then we obtain the optimal convergence order for the energy norm.The numerical results show the effectiveness of our method.For the singularly perturbed convection-diffusion problems,an upwind finite vol-ume element method on the Shishkin mesh is constructed.By using the upwind tech-nique to discretize the convective term,we get the stability of the method when the lower bound of the reaction coefficient and the coefficient of the convection term sat-isfy a certain condition.Furthermore,with the help of the decomposition theorem of the solution,we analyze the energy norm error estimation of the scheme,and obtain the optimal convergence result.This result is independent of the singular perturbation parameterε.Numerical examples are provided to illustrate our theoretical results.For the fourth-order singularly perturbed elliptic boundary value problems,we construct and analyze a Bogner-Fox-Schmit(BFS)element finite volume scheme on a Shishkin mesh.Firstly,we define the discrete H~2semi-norm and discrete H~1semi-norm,which have the aspect ratio of the rectangular element,and prove the equivalence with the corresponding continuous H~2and H~1semi-norms,where the equivalence is independent of the aspect ratio of the element.With the help of the element analysis method and the equivalence of norms,we obtain the stability of the scheme.Next,ap-plying the decomposition of solution,a special interpolation of the BFS element on the Shishkin mesh is constructed and the interpolation error estimate is given.Based on the stability analysis and the interpolation error estimation,the optimal energy norm error estimation is proved.Finally,numerical experiments verify the effectiveness of our method.