Study On The Convergence Of Finite Volume Element Methods Based On Optimal Stress Points

In this paper,a new kind of Lagrange-type finite volume element methodbased on optimal stress points is presented .In general, trial space Uh andtest space Vh are chosen as the Lagrange-type finite element space and thepiecewise constant function space respectively. It is proved that the methodhas optimal H1 and L2 error estimates. The superconvergence of numericalgradients at optimal stress points is discussed.First, we consi?der the two point boundary value problem:In the sense of generalized functions, the finite volume element schemefor problem (0-0-1) is: Find uh∈Uh such thatwhereWe define an interpolation projector then the system(0-0-2) is equivalent toThen, quadratic finite volume element method based on optimal stresspoints for problem (0-0-1) has following results, when the Uh chosen La-grangian quadratic finite element space. Theorem 0.1 For su?ciently small h, a(uh,Πh?uh) is positive defi-nite,that is,there exists a positive constantβ> 0 independent of Uh suchthatTheorem 0.2 Suppose u∈H3(I) and uh are the solutions of the prob-lem (0-0-1) and the finite volume element scheme (0-0-2) respectively, thenthe following estimate holds for su?ciently small h:Theorem 0.3 Let uh be the solution to ) with, Then the following estimate holds:Theorem 0.4 For the finite volume element scheme (0-0-2) approxi-mating the two point boundary value problem (0-0-1), the following interpo-lation weak estimate holds:Theorem 0.5 Let u∈HE1(I) be the solution of the two point boundaryvalue problem (0-0-1) and uh∈Uh of the finite volume scheme (0-0-2), Assumein addition u∈H4(I),thenCubic finite volume element method based on optimal stress points forproblem (0-0-1) has following results, when the Uh chosen Lagrangian cubicfinite element space.Theorem 0.6 For su?ciently small h, a(uh,Πh?uh) is positive defi-nite,that is,there exists a positive constantβ> 0 independent of Uh suchthat Theorem 0.7 Suppose u∈H4(I) and uh are the solutions of the prob-lem (0-0-1) and the finite volume element scheme (0-0-2) respectively, thenthe following estimate holds for su?ciently small h:Theorem 0.8 Let uh be the solution to (0-0-2), and u to (0-0-1) with, Then the following estimate holds:Theorem 0.9 For the finite volume element scheme (0-0-2) approxi-mating the two point boundary value problem (0-0-1), the following interpo-lation weak estimate holds:Theorem 0.10 Let u∈HE1(I) be the solution of the two point bound-ary value problem (0-0-1) and uh∈Uh of the finite volume scheme (0-0-2),Assume in addition u∈H5(I),thenwhere r is the number of points inAt last, consider the first boundary value of the Poisson equation:Let ? be a rectangular regionis the boundary of ?, and f∈L2(?), we denote by Th the rectangular quasi-uniform decomposition for the region ?, the nodes of dual partition Th? are 2×2optimal stress points related to rectangule, Uh chosen Lagrangian biquadraticfinite element space.The finite volume element scheme for problem (0-0-15) is: Find uh∈Uhsuch that whereWe define an interpolation projector then the system (0-0-16) is equivalent toThen,we proof the positive definite of the bilinear form a(·,Πh?·) and theconvergence estimate. We define an interpolation projectorthen the system (0-0-16) is equivalent toBiquadratic finite volume element method based on optimal stress pointsfor problem (0-0-15) has following results:Theorem 0.11 Let Th be a rectangular decomposition, Then the bilin-ear form a(·,Π?h·) is uniformly positive definite,there exists a constantα> 0independent of Uh such thatTheorem 0.12 Suppose u∈H3(?) and uh are the solutions of theproblem (0-0-15) and the biquadratic finite volume element scheme (0-0-16)respectively, then the following estimate holds :Theorem 0.13 Let uh be the solution to (0-0-16), and u to (0-0-15)with , Then the following estimate holds:Theorem 0.14 For the biquadratic finite volume element scheme (0-0-16) approximating the problem (0-0-15), the following interpolation weakestimate holds: Theorem 0.15 Let u∈H01(?) be the solution of the problem (0-0-15)and uh∈Uh of the biquadratic finite volume scheme (0-0-16), Assume inaddition u∈H4(?),thenwhere r is the number of points in...