Finite Element Method And Finite Volume Method Based On B-Spline

In this paper,finite element method and finite volume method based on linear B-spline and quadratic B-spline and cubic B-spline are introduccd.And precondition of linear B-spline wavelet is introduced. We achieve a variety of methods with Matlab.Let m be any positive integer and let N_m(x) denote the rath order B-spline.More precisely,N_m(x) is defined recursively bywith N₁(x) =χ_[0,1]-The mth order B-spline waveletwith support [0,2m-l],is a basic wavelet that generates W₀,and consequently , all the wavelet spaces W_j,k∈Z([9]).We consider the two-point boundary value problem,among them ,p∈C(I),p≥0,f∈C(I),I = [a, b].Let U_h is the finite element space ,and rth B-spline function {φ_i(x)} is a basement of U_h .Any u_h,∈U_h can be expressed as u_h(x) = (?)c_iφ_i(x).The bilinear form isGalerkin function is If r = 1,theni,j = 0,l,…,n-2. If r = 2,thcn i, j = -1,0,…,n-2. If r = 3,then i, j = -2, -1,0,…,n - 2. By ||.||₁ and ||.||, finite element methods based on the rth B-spline have convergence order of O(h^r) and O(h^r+1).Select rth B-spline functions as the basement of the trial function space and piecewise constant functions as the basement of test function spaces.We can get integral form of conservation of (3): select u∈H_E¹ = {u| u∈H¹(I), u(a) = 0, u(b)—0},satisfyingHere,J = [c_j, d_j] depends on the trial function space.Let U_h is the limited subspacc of H_E¹.The approximate form of (3a) : select u_h,∈U_h, satisfyingFinite volume methods based on linear B-spline and cubic B-spline have same convergence order with finite element method.H¹ convergence order and L² convergenceorder based on quadratic B-spline are both O(h²).Consider (3).For the initial mesh generation T_h⁽⁰⁾ : a = x₀ < X₁ <…< x_N,h = x_i - x_i-1,select linear B-spline functionΦ⁽⁰⁾ = {φ₀⁽⁰⁾ ,φ₁⁽⁰⁾,….φ_N-2⁽⁰⁾ as the basement of finite element space . Encrypt mesh generation,and let x_i+(?) = x_i + (?),i = 0,1,…,N - 1, T_h⁽¹⁾ : 0 = x₀ < x_? < x₁ <…< x_N-(?) < x_N = 1, finite element space in the new mesh generation isBasement of wavelet space W⁽⁰⁾ is There is relationWe know V⁽¹⁾ = V⁽⁰⁾ (?)W⁽⁰⁾.There is the matrix L₀ that meetsA_Φ⁽⁰⁾ is the stiffness matrix of finite element space V⁽⁰⁾ ,and A_φ⁽⁰⁾ is the coefficientmatrix after transforming basement.Then they have relation:At nth encryption ,there is the conclusion withκ(D_nG_nA_ψ⁽ⁿ⁾ G_nD_n) = O(1).D_n is a diagonal processing matrix and G_n is the standardized matrix.The boundary wavelets satisfy...