L~2 Error Estimates And Superconvergence Of High Order Finite Volume Element Methods On Triangular Meshes

The finite volume element method (FVEM) is a main discretization technique for par-tial differential equations. Due to have local integral conservation, can keep local con-servation of mass, momentum, energy and other physical quantities, the finite volume element method is widely used in fluid mechanics, electromagnetism, semiconductor simulation and petroleum engineering and other fields.The study of the finite volume element method theory for second order elliptic equation mainly includes stability, convergence and superconvergence analysis, etc. Among them, the stability of FVEM scheme (i.e. the positive definiteness of bilinear form) is the basis of error estimates.The analysis of the linear finite volume element method on triangular meshes has been well completed and clear. However, so far, there existed no published results about the L2 error estimate and superconvergence of high order FVEM schemes on triangular meshes. In particular, for the quadratic finite volume element method, people are even not sure about whether the optimal L2 convergence order can achieve or not, not to mention the existence of superconvergence.Consider the second order elliptic equation with variable coefficient as model prob-lem.1) By solving the resulted equations of the proposed orthogonal conditions, the optimal L2 convergence rate can be obtained for any k order finite volume element method on triangular meshes.2) With Aubin-Nitsche technique and orthogonal con-ditions, we prove the L2 error estimate.3) In particular, for quadratic finite volume method on triangular meshes, the scheme with optimal L2 convergence rate is also the unique quadratic scheme holding superconvergence. The results of superconvergence are proved by using unit cancellation technique and orthogonal conditions.Firstly, for Lagrange finite volume methods on triangular meshes, orthogonal con-ditions are proposed to restrict the dual partition ?h*. Given primary partition ?h and k order trial function space, the corresponding (k-1) orthogonal conditions are proposed as, Here, li(i=1,2,3) are 3 sides of K. ?h* is the piecewise constant interpolation operator based on ?h*.By solving the nonlinear equations resulted by (k-1) order orthogonal conditions (1) and (2), any k order (k> 0) FVEM scheme with optimal L2 convergence rate can be obtained. In particular, for quadratic finite volume methods, the scheme resulted by orthogonal conditions is the unique quadratic FVEM scheme which holds supercon-vergence. To be sure, orthogonal conditions are sufficient conditions to guarantee the optimal L2 convergence rate for FVM schemes.Secondly, the L2 error estimate of FVEM are studied for second order elliptic e-quation on triangular meshes. A unified framework for the proof is presented.In order to obtain the L2 error estimate of FVEM, the difference between the bi-linear form a(·,·) of FEM and the bilinear form ah(·,?h*·) of FVEM should be esti-mated. Using the piecewise linear interpolation operator ?h1 based on ?h(instead of piecewise k order interpolation operator ?hk), the difference between the bilinear form a(u-uh,?h1w) of FEM and the bilinear form ah(u-uh,?hlw)) of FVEM can be transferred into two parts:integral in the interior of element and integral on element boundaries.Then, with Aubin-Nistche technique, orthogonal conditions and the stability of FVEM schemes, the unified proof for the L2 error estimate of any k order orthogonal finite volume element method can be presented as follows (Theorem 1).Theorem 1. Suppose that u ?(?) ? Hk+2(?) is the solution of (2.1), and ?h, is regular. For k-order Lagrange trial function space Uh, choose ?h,ort* satisfying (k-1)-order orthogonal conditions. Then, there exists a positive constant C such thatFinally, the superconvergence of quadratic FVEM scheme is studied on triangular meshes. When the exact solution of (2.1) satisfies certain regularity, the primary parti-tion is C uniform and the dual partition satisfies orthogonal conditions, the correspond-ing FVEM scheme holds the weak estimate of the first type, the superconvergence with H1 norm, the superconvergence with L2 norm and the superconvergence of the FVEM solution on interpolation nodes.With orthogonal conditions and unit cancellation technique, using the results of the stability of FVEM [11,69] and the results of superconvergnece of FEM [18,65], we proved the weak estimate of the first type for quadratic orthogonal FVEM scheme (Theorem 2).Theorem 2 (The weak estimate of the first type). Let ?h be a C-uniform mesh of?. Suppose u ? H01(?) ?H4(?), then Here uI Uh is a quadratic interpolation of u in ?.Let wh=Uh-uI. Then, with the stability of FVEM [11,69] and Theorem 2, we have the superconvergence of uh-uI with H1 norm (Theorem 3).Theorem 3. For elliptic problem (2.1), let Th be a C-uniform mesh of S?.And suppose that u?H01(?)?H4(?), thenBased on Theorem 3, with Aubin-Nitsche technique and orthogonal conditions, we can obtain the superconvergence of uh-uI with L2 norm and the superconvergence of u-uh on interpolation nodes(i,e.the superconvergence with discrete L2 norm) (Theorem 4).Theorem 4.For elliptic problem(2.1),let ?h be a C-uniform mesh of?.And suppose that u ? H01(?)?H4(?),then where N denotes all interpolation nodes.?·?0,N is the discrete L2 norm.Numerical results are also presented to demonstrate the proved result.