Mixed Finite Volume Element Method

Recently, more and more researchers are interested in the mixed finite volumeelement method at home and abroad. In general, The mixed finite volume elementmethod represents the local conservation of certain physical quantities. Because ofthis natural association and its simplicity, the finite volume element method is widelyused in computational ?uid mechanics, solid mechanics, electromagnetic and otherapplications.The main work of this paper is:(1). Three-dimensional mixed finite volume element methodWe consider the following elliptic problem in a bounded convex domain Let us introduce a new variableand write the problem (0-1)-(0-2) as the system of first-order partial di?erential equa-tionsThe weak formulation is: find such that,where Let Th = {KB} be a Tetrahedron partition, KB is the Tetrahedron element withbarycenter B. Denote h_KB is the diameter of KB. We will choosethe lowest order Raviart-Thomas mixed finite element space as the trial space[36, 60]Define the Raviart-Thomas projectionwhere Si is the surface of element K.The mixed finite element method for (0-1)-(0-2) is: find such that,Now we construct the dual partition Th - related to Th. Let A₁A₂A₄ - A₅ andA₂A₃A₄ - A5 be the two tetrahedron element, denote by KB₁ and KB₂, respectively.B₁,B₂,B are the barycenter of tetrahedron KB₁,KB₂ and triangle A₂A₅A₄, re-spectively. Connect A₂B₁, A4B1, A₅B₁, A₂B₂, A₄B₂, A₅B₂ to form a hexahedronB₁ - A₂A₅A₄ - B₂, called a dual element related to SA₂A₅A₄, denote by KS-A₂A₅A₄,or KB-. All the dual elements constitute a new partition of -, called dual partition,denote by Th - = {KB-}.Next we construct the test function V h. Define the operatorγh : Uh→V h:-The mixed finite volume element method for such that,where Lemma 0.1 The following relations hold:By Lemma 0.1, (0-11)-(0-12) can be rewritten as: find suchthat,Lemma 0.2 There exits a positive constantαindependent of h such thatBy Lemma 0.1, we haveLemma 0.3 There exits a positive constantβindependent of h such thatThe Lemma 0.2 and Lemma 0.3 imply the uniqueness and existence of the solution of(0-18)-(0-19).Theorem 0.1 Let (u,p) and (u?h, p?h) be the solution of (0-6)-(0-7) and (0-18)-(0-19), then there exits a positive constant C independent of h such thatThe scheme (0-18)-(0-19) is nonsymmetric, next we give a symmetric mixed finitevolume element scheme. In fact, it is an approximation of the scheme (0-18)-(0-19). Clearly the scheme (0-23)-(0-24) is symmetric, and is simpler in computation then thescheme (0-18)-(0-19).The following theorem is obvious.Lemma 0.4 Then there exits a positive constantαindependent of h such thatBecause of Lemma 0.4 and Lemma 0.3, the scheme (0-23)-(0-24) have a unique solution.Moreover, the following convergence result holds:Theorem 0.2 Let (u,p) and (uˉh, pˉh) be the solution of (0-6)-(0-7) and (0-23)-(0-24), then there exits a positive constant C independent of h such thatprovided that(2). Superconvergence of mixed covolume method on quadrilateral gridsfor elliptic problemsWe consider the following second-order elliptic problem in a bounded convex polyg-onal domainLet us introduce a new variable u = ?K p and write the problem (0-27)-(0-28)as the system of first-order partial differential equationsequipped with the norm We adopt V×W as the weak solution spaces, whereThen the associated weak formulation of (0-29)-(0-31) is to seek a pair (u,p)∈V×Wsuch that Difine the Piola transformationWe say that the quadrilateral partition Th satisfies almost parallelogram conditionif each element Q∈Th is h2-parallelogram, i.e.,We will choose the lowest order Raviart-Thomas mixed finite element space V h×Wh (?) V×W[36, 60] as the trial spacewhere the local space V h(Q ?) is defined to beDenoting the nodal basis forthen the nodal basis for V h is given by?The Raviart-Thomas projectionΠh[36] is defined as follows:whereΠ|- is defined to beDefine the piecewise constant interpolant Qhp[107] aswhere cQ is the center of element Q. Next we define the test function Y h×Wh, with Wh defined as before. Define thetransfer operator is defined in a similar way. The velocity test space Y h is defined to be spannedby them. The transfer operatorγ_h : V_h→Y _h is then defined as follows: If v_h∈V_his expressed in the formthen we set Now the mixed covolume method for the problem (0-27)-(0-28) is as follows: FindNext we give two useful lemmas.Lemma 0.5 Assume that the partition Th satisfies the almost parallelogramcondition, then for every we haveLemma 0.6 There exists a positive constant C independent of h such thatBy the above two lemmas we have easy access to the following superconvergence result.Theorem 0.3 Assume that the partition Th satisfies the almost parallelogram(3). A stabilized mixed finite volume element method on rectangulargrids for the Stokes equationsWe consider the following stationary Stokes problem in an axiparallel domain We adopt V×W as the weak solution spaces, then the associated weak formulationof (0-49)-(0-51) is to seek a pair (u,p)∈V×W such thatWe choose the lowest equal-order finite element space V h×Wh ? V×W as thetrial space:Now, we can define the bilinear form G(·,·) as follows:Then the stabilized finite element method for the Stokes problem (0-49)-(0-51) is toseek (uh,ph)∈V h×Wh such thatwhereLet be the barycenter dual partition corresponding to Th. Define the following twotest spaces: In addition, define two operatorswhereχPj is the characteristic function of the dual elementDefine the following bilinear forms of the finite volume method as follows:To define the stabilized finite volume method, we need add a stabilization term tothe variational formulation associated with the Stokes equations with the idea of [32].Define the following bilinear formNow we definethen the new stabilized finite volume method for the Stokes problem (0-49)-(0-51) is:The new stabilized mixed finite volume element method is stable.Theorem 0.4 The following holdand, there exists a positive constantβindependent of h such that We have the following superconvergence results.Theorem 0.5 Let (u,p) and (u?h, p?h) be the solution of the Stokes problem(0-53) and the stabilized finite volume system (0-71), respectively. Then(4). Superconvergence of mixed finite element method on rectangulargrids for the Stokes equationsUsing a technique similar to that in [17], We discuss the superconvergence of mixedfinite element method (0-59). Fist, we give some useful lemmas.From the previous few lemma, we have easy access to the following superconver-gence result.