Inite Volume Element Methods Based On Constrained Nonconforming Rotated Q1Elements

Finite volume method (FVM) is an important numerical discretization technique for solving partial differential equations, especially for those arising from physical conserva-tion laws including mass, momentum and energy. In general, FVM has both simplic-ity in implementation and local conservation property, so it has enjoyed great popular-ity in many fields, such as computational fluid dynamics, computational aerodynamics, petroleum engineering and so on. About some recent development of FVM, readers can refer to the monographs [46,49,61,79,88,92,93,112,118] for details.In2003, Park and Sheen [98] have introduced a P1-nonconforming quadrilateral element which has only three degrees of freedom on each quadrilateral. Later, Man and Shi [93] proposed the P1-nonconforming quadrilateral FVM for the elliptic problem by using a dual partition of overlapping type. Following the line of the finite element in [98], In2005, Hu and Shi [73] presented a constrained nonconforming rotated Q1(CNRQ1) element and applied it to the second order elliptic problem. In [73], the authors also point out that the CNRQ1element and the P1-nonconforming element are equivalent on a rectangle since the constraint and the continuity are the same, however, the two elements are different on a general quadrilateral. Afterwards, Hu, Man and Shi [72] and Mao and Chen [94] investigated and analyzed the CNRQ1-P0finite element method for the Stokes problem on rectangular meshes. The application of the CNRQ1element to the nearly incompressible planar elasticity problem can be referred to [72,95]. Meanwhile, Mao and Chen [94] and Liu and Yan [90] discussed the supconvergence of the finite element scheme for the Stokes problem on rectangular meshes.The main work of this paper is as follows:1. Finite volume element method for the Stokes problems.Let Ω be a bounded, convex and open polygon of R2with boundary (?)Q. We consider the following Stokes equations with the homogeneous Dirichlet boundary condition-△u+▽p=f, in Ω (9) divu=0, in Ω,(10) u=0, on (?)Ω,(11) where u=(u1,u2) represents the velocity vector, p is the pressure, and f indicates a prescribed body force.Let Th={K} which is called primal partition be a strictly convex and quadrilateral partition of Ω. The dual partition is nonoverlapping barycenter quadrilateral partition. Choose CNRQ1function space Uh as the trial function space and the piecewise constant space Vh as the test function space for the velocity. For the pressure, we select the piecewise constant space(Ph) on the macro-elements as discrete space. The FVEM for the Stokes problem (2.1)-(2.3) investigated in this paper is:find a pair (uh,ph)∈Uh×Ph, such that ah(uh,vh)+bh(vh,ph)=(f,vh),(?)vh∈Vh,(12) b'h(uh,qh)=0,(?)qh∈Ph.(13)(1). Stability.Suppose the primal partition is quasi-uniform and regular. Each quadrilateral in Th satisfies quasi-parallel quadrilateral condition. Then, we obtain the following results:Lemma1. For sufficiently small h, there exists a constant C0>0, we have ah(uh,Γhuh)≥C0|uh|1,h2,(?)uh∈Uh.Lemma2. The pair (Uh,Ph) satisfies an uniform inf-sup condition, i.e., there holds(2). Error estimates.We prove the errors in the H1norm and L2norm for the velocity, the error in the L2norm for the pressure, and the superconvergence of the velocity in the broken H1norm and the pressure in the L2norm.. Theorem1. Let the pair (u,p)∈(H01(Ω)∩H2(Ω))2×(L02(Ω)∩H1(Ω)) be the solution of (9)-(11); and (uh,Ph)∈Gh×be the solution of (12)-(13). There exists a positive constant C independent of h such that|u-uh|1,h+||p-Ph||0≤Ch(||u||2+||p||1).Theorem2. Let (u,p)∈(H01(Ω)∩H2(Ω))2×(L02(Ω)∩H1(Ω)) and (uh,ph)∈Uh×P be the solutions of (9)-(11) and (12)-(13), respectively. If f∈H1(Ω)2, then||u-uh|0≤Ch2(||u||2+||p||1+||f||1).Theorem3. Let (u,p)∈(H01(Ω)∩H3(Ω))2×(L02(Ω)∩H1(Ω)) and (uh,ph)∈Uh×Ph be the solutions of (9)-(11) and (12)-(13), respectively. There exists a positive constant C independent of h such that|Πhu-uh|1,h+||Jhp-Ph||0≤Ch2(||u||3+||p||2).(15)Let Π2hvh∈Q2(T) and J2/1be the postprocessing interpolation operators: where ψi,i=1,2,…,8be the associated basis functions of the biquadratic element space Q2(τ), Q1(τ) is the bilinear function on the element τ. Then we have:Theorem4. Suppose all condition of the thmm above are valid,|Π2huh-u|1+||J2hPh-p||o≤Ch2(||u||3+||p||2).1. Finite volume element method for the linear elasticity problem.Consider the pure displacement boundary value problem for the equations of plain strain linear isotropic elasticity: where u denotes the displacements, f the body forces and μ,λ are Lame parameters.Let Th={K} which is called primal partition be a strictly convex and quadrilateral partition of Ω. The dual partition is nonoverlapping barycenter quadrilateral partition. Choose CNRQ1function space Uh as the trial function space and the piecewise constant space Vh as the test function space for the displacements. The FVEM for the linear elasticity problem is:Suppose the primal partition is quasi-uniform and regular. Each quadrilateral in Th satisfies quasi-parallel quadrilateral condition.Then,we can obtain the following results to prove the FVEM is locking-free.(1).Error estimate in energy norm. Theorem5.Suppose Uh∈Uh and u∈H01(Ь)∩H2(Ω) are solutions of finite uolume element scheme(19)and the equations(18),respectively,then||u-uh||h≤Ch||f||0Ω(2).Superconvergence in the energy norm.Theorem6.Suppose Uh∈Uh and u∈H01(Ω)∩H3(Ω) are solutions of finite uolume element scheme(19)and the equations(18),respectively,then||Πu-uh||h≤Ch2(||u||3,Ω+λ||divu||2,Ω).