Some Researches Of High Order Finite Volume Element Method For Second Order Elliptic Equation

The finite volume element method, also called as generalized difference method, was firstly put up by professor Ronghua Li in 1982. Due to its computational simplicity and preserving local conservation of certain physical quantities, it has been widely used in computing fluid mechanics, solid mechanics, electromagnetic field and other fields. At present the schemes and the theoretical results of the finite volume element method are mostly about low order element, but that for the high order element are less.The main work of this paper is:(1). The discussion of the symmetry and the L2-norm convergence rate of Lagrange quadratic finite volume element methodSupposeΩ, (?) R2 is a bounded convex polygon with boundaryΓ=(?)Ω, and consider the Poisson equation: where f∈L2(Ω).The corresponding variational problem of Equation (0-1) is:find u∈H01(Ω) such that whereLet Th be a finite element partition ofΩinto "perfect" regular triangulation. Denote Th={KQ}, Q is the barycenter of the triangle. The vertexes of the triangles and the midpoints of the sides are taken as the nodes. LetΩh be the set of nodes of Th,Ωh=Ωh\Γthe set of the inner nodes, Mh be the set of the midpoints of the sides of all the elements and Mh=Mh\(?)Ω.The trial space Uh is chosen as the Lagrangian quadratic element space related to the triangulation Th.The dual partition Th* related to Th is consisted of the polygons KP0* surrounding the node P0∈Ωh and KM* surrounding M∈Mh.Their detailed construction is as follows:(1) Construction of KP0*.Suppose that P0∈Ωh, that Pi(i=1,2,…,6) are its adjacent vertexes, and that P0i is a point on P0Pi such that P0P0i/P0Pi=αandα∈(0,1/2). Connect successively P0i(i=1,2,…,6) to obtain a polygon KP0* surrounding P0.(2) Construction of KM*.Let M∈Mh be a midpoint of a common side of two adjacent triangular elements KQ1=ΔP0P1P2 and KQ2=ΔP0P2P3. Denote by M12,M23,M30, M01 the midpoints of P1P2, P2P3, P3P0, P0P1 respectively. And let Q12,Q23,Q03,Q01 be the points satis-fied A polygon KM* surrounding M is obtained by connecting successively P20, Q03, Q2, Q23, P02, Q12, Q1,Q01,P20.So we can obtain the different dual partition when choosing differentα,β. In generally, we requireα≤β. The paper [75] chooseα=β=1/3, and the paper [49] chooseα=1/4,β=1/3.The test function space Vh is taken as the piecewise constant function space related to the dual partition Th*.Choose the Uh and Vh defined above, the Lagrange quadratic finite volume element method of the problem (0-1) is:find uh∈Uh such that whereWe can get the element stiffness matrix by computing the equation (0-4) on dual element of KQ and see that (α,β) which makes the element stiffness matrix symmet-rical is not exist, i.e. under the present subdivision of this dual partition within the framework, the symmetry of the method can't be obtained. But when we choose the right angled isosceles triangulation, the element stiffness matrixIf we chooseα=β=1/2-(?)/6, A1 and A4 is symmetrical respectively, but (α,β) which makes A2 symmetrical to A3 is not exist.when we choose the equilateral triangulation, the element stiffness matrix when (α,β)∈(0,1/2)×(0,2/3], B1,B2,B3,B4 is symmetrical respectively, but (α,β) which makes B2 symmetrical to B3 is not exist either.By the numerical experiments, we can see that the H1-norm convergence rates of the Lagrange quadratic finite volume element method are always O(h2) under the different dual partitions. But the L2-norm convergence rates become different when the different dual partitions are chosen. Under some dual partitions, the L2-norm con-vergence rates are shown to be faster than the H1-norm convergence rates, sometimes even to be O(h3); but for some dual partitions the two convergence rates are shown to be the same.(2). Finite volume element method with Lagrangian cubic functionsLet Th be a finite element partition ofΩinto "perfect" regular triangulation, then we obtain the primary partition Th, where h is the maximum length of all triangles' sides. Denote the triangle element in Th by KQ, Q being the barycenter of the triangle element.The vertexes Pl, the trisection points Pl,l+1,Pl+1,l(l=i,j,k; i+1=j,j+1= k, k+1=i) and the barycenters Q are taken as the nodes. LetΩh be the set of nodes of Th andΩh=Ωh\Γthe set of the inner nodes.Now we define the dual partition Th* related to Th on the refinement partition Th/3. Here we choose the barycenter dual partition used in Lagrangian linear finite volume element methods. Here, we always assume that Th (or Th/3) and Th* are regular([55]).The trial function space Uh is chosen as the Lagrangian cubic element space related to Th. The test function space Vh is taken as the piecewise constant function space related to the dual partition Th*.Then for Equation(0-1),the Lagrangian cubic finite volume element scheme is: find uh∈Uh such that whereDefineΠh*uh to be the interpolation of uh∈Uh onto Vh,then the finite volume element method reads:find uh∈Uh such thatNow let the vector z:=[z1,z2,…,z9]T be defined byIntegrate the left hand of(0-8)on KQ to get: where Here denote by K the set of the ten nodes of KQ=ΔPiPjPk.we assume that the three interior angles areθi,θj,θk corresponding vertexes.For simplicity,we letαl=cotθl(l=i,j,k).Finally,we can obtain the unit quadratic form by computation Then we symmetrize (0-12) results in where B=A+AT/2 is a 9×9 symmetrical matrix.Here, M～N means M(?)N and N(?)M, where M(?)N means that M can be bounded by some constant multiple of N where the constant is independent of the various parameters the quantities M and N may depend on.Without loss of generality, we always suppose thatθi≤θj≤θk, then 0<αj≤αi andαk≤αj≤αi. Noticing that Th is a regular triangulation, so for all l=i,j,k, we haveθl≥θ0, whereθ0 is called the minimal angle.Lemma 0.1 For all uh∈Uh, there holds:Then by Equation (0-15), we haveLemma 0.2 If B is uniformly positive definite, that is then the inequality holds for all KQ∈Th, where a is a positive constant.By the above lemma0.2, the proof of Equation (0-18) is reduced to the proof of the positive definiteness of B. We have:Theorem 0.1 For all KQ∈Th, let B be the matrix defined above. If the minimal angleθ0≥17.25°, then the positive-definiteness Equation (0-17) holds, i.e. Equation (0-18) holds.Because of the positive definiteness Theorem0.1, the scheme (0-6) have a unique solution. Moreover, we have the convergence result: Theorem 0.2 Suppose that the triangulation Th satisfies the conditions in The-orem0.1. Let u be the solution to Equation (0-2) and uh to the cubic finite volume element scheme Equation (0-6). If u∈H4(Ω), then the following error estimate holds for sufficiently small h:Some numerical experiments confirm our theoretical considerations and show that the convergence rate of the maximum norm and L2-norm are O(h4). In addition, we find that the solutions of the average gradient of finite volume element method with the Lagrangian cubic function for Poisson equation has superconvergence phenomenon:Remark 0.1 The superconvergence rate of the average gradient is approxi-mately one order higher than that of the H1 norm at the symmetrical points of trian-gular element (i.e. trilateral center points and three angular points).Here the solutions of the average gradient is obtained by the following module: where Sh is the set of the symmetrical points of triangular element,▽is the average value of gradient▽and r is the number of points in Sh.(3). Hybirdization Lagrange cubic finite volume element methodWe borrow the idea of [17] which is combining the cubic finite volume element method and the cubic finite element method and give the hybirdization Lagrange cubic finite volume element method of the problem (0-1).The vertexes, the trisection points and the barycenters are taken as the nodes. Then we can obtain the original partition Th. LetΩh be the set of nodes of Th,Ωh=Ωh\Γthe set of the inner nodes,ΩhV be the inner vertexes,ΩhB be the inner barycenters andΩhT be the inner trisection points.The main idea of hybirdization Lagrange cubic finite volume element method is that we use the finite volume element method at the inner vertexes Pi∈ΩhV and use the finite element method at the inner barycenters Q∈ΩhB and the inner trisection points We define the dual partition and we choose the barycenter dual partition used in Lagrangian linear finite volume element methods. The trial function space Uh is chosen as the Lagrangian cubic element space related to Th. The test function space Vh related to the finite volume element method is taken as the piecewise constant function space related to the dual partitionFor simplicity, we denote uh on KQ: where ci is the value of uh at ten nodes andφi is the basis functions at ten nodes.Use the trial function space Uh and the test function space Vh defined above, the the hybirdization Lagrange cubic finite volume element method of the Poisson equation (0-1) reads:find uh∈Uh such that where If the triangular partitions are restricted under the condition of similar finite families, the analysis of the positive-definiteness of this method could imitated by the analysis method used in [17]. But this restriction is more stringent. Similarly, we can obtain the H1 error estimate by using the positive-definiteness result.Some numerical experiments confirm the effectiveness of our method and show that the convergence rate of the maximum norm and L2-norm are O(h4). We also find that the solution of the average gradient has superconvergence phenomenon Remark (0.1) at the symmetrical points of triangular element.(4). Extrapolation for the bilinear finite volume element methodWe choose the uniform rectangular partition, the trial space Uh is chosen as the bilinear element space related to the original partition Th and the test space Vh is chosen as the constant function space related to the dual partition Th*. Then we can construct the bilinear finite volume element method for problem (0-1), and for the finite volume solution uh, we introduce the following extrapolation algorithm ([64]):(1) On the uniform rectangular partition Th and its' refinement partition Th/2, we can compute finite volume solutions uh and uh/2;(2) Compute whereΠ3h/23 andΠ3h3 is interpolation post-processing operator, i.e. we combine the 9 small elements which is adjacent to each other into a large element K, and where P33 is the bicubic polynomial space, and zi(i=1,2,…,16) are the nodes of small element K. In order to obtain our asymptotic expansions,we have the following lemmas firstly:Lemma 0.3 For w∈H2(K),v∈P1,1(K),it exists constants bij(1≤i,j≤2), such that where B=(bij)2×2,bij=λiλjbij*,1≤i,j≤2,且whereLemma 0.4 For w∈H4(K),v∈P11(K),we have We can get the following proposition by the above lemmas:Proposition 0.1 We denote the bilinear forms on element K by a(·,·)K,ah(·,·)K,and assume that u∈H4(K),f∈H2(K),then for v∈P11(K): Then we sum all K∈Th and acquire the following basic expansion:Theorem 0.3 Suppose thatΩis a convex polygonal domain,the rectangular partition is uniform in the interior subdomainΩρ={x∈Ω,dist(x,(?)Ω)≥ρ>0}. Assume u∈H4(Ωρ)∩H01(Ω), f∈H2(Ω),then where rh(f,u,v)is the bounded linear functional in Uh,and where QD4u are some combinations of u and its partial derivatives of order less than or equal to four.So, we have the following asymptotic expansions:Theorem 0.4 If the assumptions is established occurred in Theorem0.3, then there exists w∈H4(Ωρ)∩H01(Ω) such that Finally, we haveTheorem 0.5 Suppose thatΩ, is a convex polygonal domain, the rectangular partition is uniform in the interior subdomainΩρ={x∈Ω,dist(x,(?)Ω)≥ρ>0}. Suppose f∈H2(Ω), the solution u∈H4(Ωρ)∩H01(Ω) of problem (0-2), and uh is the interpolation extrapolation solution defined in (0-27), then...