Biquadratic Element Finite Volume Methods Based On Optimal Stress Points

The finite volume element (FVE) method (also called generalized difference method, colume method, box scheme), is a discretization technique for partial differential equa-tions, especially for computational fluid mechanics, electromagnetics and semiconduc-tor simulation problems, as a result of those that arise from physical conservation laws including mass, momentum and energy. The FVE method is based on the Petrov-Galerkin variational principle:define the primal partition Th and the dual partition T*h construct the trial space Uh and test space Vh with respect to Th and T*h, re-spectively; by the Green formula, an area integral formulation of the equation on the control volume is turned to the line integral formulation along the boundary of the dual unit. Generally speaking. The FVE method is a numerical technique that lies some-where between finite element and finite difference methods; it has a flexibility similar to that of finite element methods for handling complicated solution domain geometries and boundary conditions; and it has a simplicity for implementation comparable to fi-nite difference methods. More importantly, the FVE method possesses the important property of inheriting the physical conservation laws of the original equations locally.The main difference between the FVE method is that the FVE method uses two spaces:the trial space of piecewise polynomial functions over the primal partition and the test space of lower order piecewise polynomial functions over the dual partition. In1982, R. H. Li [3-5] gave a general way to choose the test function space, tak-ing the common terms of the local Taylor expansions as the basis functions over the dual partition, and rewrote the integral interpolation method in a form of generalized Galerkin method, and thus obtained a difference method on irregular networks, that is, the so-called generalized difference methods (GDM). Since then, many domestic scholars carried out in-depth and extensive research on the theory and application of GDM, including the construction of GDM and the corresponding theoretical analy-sis for elliptic, parabolic and hyperbolic equations. Therefore, a comprehensive and systematic theoretical framework is established, most of these results have been sum-marized in the monograph [6,7]. By the approximation theory, we know that the numerical derivatives limited by the degree κ of the approximate polynomials can ob- tain only k-th order accuracy, in general this estimate can not be improved even if thesolution possesses an higher smoothness. But this fact does not exclude the possibilitythat the approximation of derivatives may be of higher order accuracy at some specialpoints, called optimal stress points. The FVE method based on optimal stress pointsfor solving partial diferential equations has been studied [8–10].For two-point boundary value problems, In [11], Xiang developed a quadratic finitevolume element method for solving two-point boundary value problems and obtainedoptimal H1error estimate, choosing trial and test spaces as the biquadratic finiteelement space and piecewise constant function space, respectively. the dual partitionratio of the method is1:2:1, i.e., each edge of an element in the primal partition hispartitioned into three segments so that the ratio of these segments is1:2:1. The authors[8] constructed a new one-dimensional high order finite volume method. Although theconstructions of the dual mesh and the test function space are complex, The diferencebetween the finite volume method and the corresponding bilinear form of finite elementmethod is a small amount. Therefore, the some optimal order error estimates of FVEmethod are obtained by means of finite element method, and the optimal stress points(Gauss points) are used as the dual element nodes in this article. In [9], a new quadraticFVE method for solving two-point boundary value problems is presented. The formof dual partition is diferent from [11], the optimal stress points (second order Gausspoints) are used as the dual element nodes, and the trial and test spaces are chosen asthe quadratic finite element space and piecewise constant function space, respectively.It is proved that the new method has optimal order H1and L2convergence order. Thesuperconvergence of numerical gradients at optimal stress points is also discussed.The numerical results are shown that the L2-norm convergence order for thequadratic FVE method in [11] is not optimal, only has the same convergence orderas H1-norm, and the superconvergence of numerical gradients at optimal stress pointsis not exist. However, the L2-norm convergence order in [9] is optimal, and has the morea convergence order than H1-norm, and the superconvergence of numerical gradientsat optimal stress points for this quadratic FVE method is exist. In [1], a biquadraticFVE method for solving elliptic problems and obtained optimal H1error estimate,the form of dual partition is the same as [11]. In this thesis, Using four interpolationoptimal stress points on every rectangle element to construct a dual partition related to primal partition, we firstly developed a new class of biquadratic FVE method forPoisson equations, after that the stability and convergence analysis for new methodare obtained, including the H1-norm, L2-norm and the superconvergence analysis atoptimal stress points, then we extend the new method to second dimensional parabolicand second order hyperbolic equations. As regards the error estimates of the semi-discrete and full discrete biquadratic FVE methods for the second order parabolic andhyperbolic problems, we can borrow the theories and techniques of finite element meth-ods to get basically parallel results. But there are certain difculties requiring specialtreatments, such as the asymmetry of (·, Π h·) and a(·, Πh·). The innovative aspects ofthis thesis are as follows(1) We put forward a new biquadratic FVE method for solving poisson problemsby using four interpolation optimal stress points on every rectangle element toconstruct a dual partition related to primal partition, choosing the trial and testspaces as the biquadratic finite element space and piecewise constant functionspace, respectively. It is proved that the new method has optimal order H1andL2error estimate. The superconvergence of numerical derivatives at optimal stresspoints is also discussed. Finally, the numerical experiment shows the results oftheoretical analysis.(2) Utilizing the constructed idea of the new method for elliptic problems, we studythe second order parabolic problems, and propose the new semi-discrete and fulldiscrete biquadratic FVE schemes. Then, we treat the asymmetry of (·, Π h·) anda(·, Π h·), and obtain the convergence analysis for the two methods respectively,including the H1-norm, L2-norm and the superconvergence analysis at optimalstress points. Finally, we compare the results of the numerical experiment byusing the new method and the existing methods, conclude that the new methodhas higher accuracy, and verify the results of theoretical analysis.(3) We extend the new method to second order hyperbolic problems, and constructa new full discrete biquadratic FVE scheme. It is proved that the new methodhas optimal order L∞(H1) and L∞(L2) error estimate. The superconvergenceof numerical derivatives at optimal stress points is also discussed. Finally, thenumerical experiment shows the higher accuracy for the new method.