Discontinuous Finite Volume Element Methods Applied To Elliptic Equations And The Stokes Equations

Discontinuous Finite Volume Element Methods Applied to Elliptic Equations and the Stokes EquationsThe Finite Volume Element Method,also called as Generalized Difference Method. Its primary merits include computational simplicity and preserving local conservation of certain physical quantities. The discontinuous finite volume element method (DFV) does not require continuity of the approximation functions across the interelement boundary but instead enforces the connection between elements by adding a penalty term.Because of the use of discontinuous functions, discontinuous finite volume element method has the advantages of a high order of accuracy, high parallelizability, localizability, and easy handling of complicated geometries.(1).Elliptic EquationsWe consider elliptic equations:whereΩis a bounded convex polygon in R²is a symmetric matrix value function and satisfies the following condition: there exists a constantβ>0, such thatβξ^Tξ≤ξ^TBξ,(?)ξ∈R².Let R_h be a triangulation ofΩwith diam(K)≤h,(?)K∈R_h.Assume that the triangulation R_h is quasi-uniform. We define the dual partition T_h of R_h for the test function space as follows. We divide each K∈R_h into three triangles by connecting the barycenter and the three corners of the triangle . Let T_h consist of all these triangles T_j.We define the finite dimensional space associated with R_h for the trial functions as V_h={v∈L₂(Ω):v|_K∈P₁(K),(?)K∈R_h}, and define the finite dimensional space P_h, for test functions associated with the dual partition T_h as P_h={q∈L²(Ω):q|_T∈P₀(T),(?)T∈T_h},where P_l consists of all the polynomials with degree less than or equal to l.Our discontinuous finite volume element scheme is to find u_h∈V_h such thatwhereα>0 is a real number that will be determined later.Numerical experiments validate that numerical solution of discontinuous finite volume element method has a first order convergence order in a discrete norm and a second order convergence order in a L² norm.(2).The Stokes EquationsWe consider the Stokes equations :whereΩis a bounded convex polygon in R², and f(x) is the external volumetric force acting on the fluid at x. We assume v=1.Let R_h be a triangular or rectangular partition ofΩwith diam(K)h. The triangles or rectangles in R_h are divided into three or four subtriangles by connecting the barycenter of the triangle or the center of the rectangles to their corner nodes. Then we define the dual partition T_h of the primal partition R_h to be the union of the triangles for both rectangular and triangular mesh. We define the finite dimensional trial function space for velocity on triangular partition by V_h={v∈L²(Ω)²:v|_K∈P₁(K)²,(?)K∈R_h},and on rectangular partition by Y_h={v∈L²(Ω)²: v|_K∈(?)(K)²,(?)K∈R_h},where (?)(K) denotes the space of functions of the form a+bx+cy+d(x²-y²) on K. Define the finite dimensional test function space W_h for velocity associated with the dual partition T_h as W_h={ξ∈L²(Ω)²:ξ|_T∈P₀(T)²,(?)T∈T_h}.Let Q_h be the finite dimensional space for pressure Q_h={q∈L₀²(Ω):q|_K∈P₀(K),(?)K∈R_h}.Let T_j∈T_h(j=1,2,···,t) be the triangles inK∈R_h,where t=3 for triangular mesh and t = 4 for rectangular mesh.The corresponding discontinuous finite volume element scheme: seeks (u_h,p_h)∈V_h×Q_h such thatwhereα>0 is a real number that will be determined later,t=3 or t=4.2)Let R_h be a rectangular partitionΩwith diam(K)≤h,(?)K∈R_h that pressure space is associated with. We divide each rectangle into four smaller rectangles by joining the opposite midsides. These smaller rectangles with size (?) generate R_?. Thetrial function space for velocity is based on the mesh R_?.Then the rectangles in R_? are divided into four subrectangles by connecting the center of the rectangles to their midpoints of each side. We define the dual partition T_h of the primal partition R_?to be the union of the rectangles with size (?). We define the finite dimensional trial function space for velocity on rectangular partition by V_h={v∈L²(Ω)²: v |_K∈Q₁(K)²,(?)K∈R_?,where Q₁(K) denotes the space of bilinear functions on K. Define the finite dimensional test function space W_h for velocity associated with the dual partition T_h asW_h={ξ∈L²(Ω)²:ξ|_T∈P₀(T)²,(?)T∈T_h}.Let Q_h be the finite dimensional space for pressure Q_h,={q∈L₀²(Ω):q |_K∈P₀(K),(?)K∈R_h}. The corresponding discontinuous finite volume element scheme: seeks (u_h,p_h)∈V_h×Q_h,such thatwhereα> 0 is a real number that will be determined later.Numerical experiments validate that numerical solutions of these two discontinuous finite volume element schemes have the same error estimate. Velocity solution has a first order convergence order in a discrete norm and a second order convergence order in a L² norm, and pressure solution has a first order convergence order in a L² norm.Numerical experiments show that discontinuous finite volume element methods applied to elliptic equations and the Stokes equations are convergent and convergence depends on the parameterαbefore the penalty term.