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Finite volume element method, also known as the Generalized Difference Method Since 1982, Professor Li Ronghua, due to its less computation, the program is easy to implement, and be able to maintain the partial conservation of physical quantities in computational fluid dynamics, solid mechanics and electromagnetic fields and other fields have a wide range of applications. This paper first introduces a the four solving one-dimensional elliptic problems the high order finite volume element method, which is based on the local L2 projection method (two methods) method based on a special mapping from Ch0 piecewise constant (two methods) . Let Ω = (xL, xR) is bounded interval, consider the two-point boundary value problem: Here, a, β, γ, f are defined in Ω real-valued function, and now, we are in the subinterval ω = ( ωL, ωR) on points, can be obtained particularly when ωR = xR, (1b) can be obtained if uh bounded subinterval (ωL, ωR) meet (2a), we have referred to as first The type Ⅰ bounded control element. Accordingly, if uh meet on a bounded subinterval (ωL, ωR) (2b), claimed that it is the first type Ⅱ bounded control volume element. The basic factors of our numerical method is a linear operator (?) H: in Phr → L2 (Ω), and satisfies the following stability consistency assumption: issue (1a) - (1b) of the discrete variational form Description can be defined as follows: h ∈ (0,1), find uh ∈ Shr = Phr ∩ H (Ω), st here bilinear form Bh: Hh2 × Phr → R defined as given some local conservation methods, they Shr from Pr to the PR-2 (R ≥ 2), or from PR to PR-1 (r ≥ 2 and is an odd number), the local L2 projection operator method. Proposition 1 Let r ≥ 2 Λh: Phr → Phr-2 is defined as follows: if: (1) Λh: When α = r-1 (3) was established when σ = 0, (4). (2) method (5) is dual unit and local conservation method, here IJhh both type Ⅰ and type Ⅱ bounded control. : (3) solution uh belongs Shr ∩ C1 (Ω) satisfy the homogeneous Neumann condition and in xR at. Proposition 2 let m ∈ N, r = 2m, (?): Pnr → Phr-1 defined as follows: if: (1) Λn: When α = r (3) established when σ = 1 (4) establishment of . (2) method (5) Shr for the volume control on the local conservation here range IJhh of type Ⅰ bounded control. Ie: we define a piecewise constant function to from fragmentation on ChO continuous mapping special mapping, and use it to derive Shr on the finite volume method (r ≥ 2) Let sR [0,1] partition a node, i.e., (?) 0 = No 0, (?) s1, (?) s-1 Lt.; (?) s, 1 j = 1, ..., s. (2) The method (5) is a dual unit as {{(xj-1H (?) I-1, xj - h (?) I)} i = 1R = 1} j = 1JH and local conservation method, where The range is the first type and the second type is bounded control element. : (3) solution uh belongs Shr ∩ C1 (Ω) satisfy the homogeneous Neumann condition and in xR at. ) established when σ = 1 (4) established. (2) method (5) Shr for the volume control on the local conservation here range IJhh of is the type Ⅰ bounded control element.: If r ∈ {2,4,6} and the method (5) following dual unit finite volume method, where, after an interval first type Ⅱ bounded controls, IJhh is the type Ⅰ bounded control yuan. consider Poisson boundary value problem: Let Ω is a rectangular region Ω = {( x, y) | a ≤ x ≤ b, c ≤ y ≤ d}, γ = (?) Ω of Ω boundary, f ∈ L2 (Ω). to Ω the uniform rectangular subdivision Th node collection ωH = {( xi, yj) | i = 0,1, ..., and M; j = 0,1, ..., N}, each small rectangle is called the unit, referred to as K. [1] directly generalized to the two-dimensional case of the discrete variational formulation of the problem (6) described can be defined as follows: h ∈ (0,1), find uh ∈ shr Shr st here bilinear Form B is defined as Qh = (?) h × (?) h which has been tried and found that does not work. Final draw text [1] even split of practice, method and four extended to two-dimensional rectangular online product shaped secondary finite volume method (Generalized Difference Method). To do uniform rectangle Sectioning TH, to Ω node collection ωH = {(xi, yj) | i = 0,1, ..., and M; j = 0,1, ..., N}, each of the small rectangular called unit, denoted by for K, HX, HY, respectively, for the x, y direction step. h = max {hx, hy} side of the midpoint of all rectangular unit is denoted as Mh, the the of all rectangular unit center of gravity is denoted as Qh, Mh = Mh \\ (?) Ω, Ωh = Ωh \\ (?) Ω. again do Th, dual split, denoted by Th *, it by surrounding nodes M ∈ Mh rectangular domain KM * P ∈ Ωh's is rectangular domain, Kp * Q ∈ Qh rectangular domain KQ * composition. Trial function the space Uh taken as corresponding to Th the product shaped lagrange secondary finite element space. Test function space Vh is taken as piecewise constant space corresponding to the dual subdivisions Th *. Let us define the format of the finite volume method by the method a promotion and methods. Seek uh ∈ Uh makes: or equivalent. (?) KQ * (?) KM * (?) KP * is the dual unit KQ *, KM * KP * boundary. Found by numerical experiments, the finite volume method by a promotion is unsolvable, promotion Method Four finite volume method has the best L2 norm error convergence order.
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