The Weighted Function Of Algebra Splines In WEB Method

The finite element method (FEM) has become the most widely accepted general pur-pose technique for numerical simulations in engineering and applied mathematics.IN the standard finite element method,constructing good meshes can be rather time-consuming.In particular, in higher dimensions, grid generation is often a bottleneck in finite element sim-ulations;Moreover, on unstructured meshes,the lack of smoothness leads to a dramatic in-crease in the dimension of Vh for higher degree approximations. Finally, the accuracy near the boundary is poor unless isoparametric elements are used.Hollig's WEB finite element method solves these problem well,and satisfies the boundary conditions by using the so-called weighted functions.WEB FEMB-Splines The uniform B-spline bn of degree n is defined by the recursion starting from the characteristic function b0 of the unit interval [0,1).Tensor production B-splines:The m-variate b-spline Bk,hn with degree nv in the vth vari-able,index k=(k1,…, km),and grid width h is defined asSplines on bounded domain:The splines Bhn(D) on a bounded domain D (?)Rm consist of all linear combinations of relevant B-splines; i.e.,the set K of relevant indices contains all k with bk,hn≠0 for some x∈D Inner and outer B-splines:We partition the grid cells Q= lh+[0, 1]mh into interior,boundary,and exterior cells,depending on whether D c D,the interior of Q intersects (?)D,or Q∩D is empty. Among the relevant B-splines bk, k∈K,we distinguish between inner B-splines bi, i∈I,which have at least one interior cell Qi in their support,and outer B-splines bj, j∈J= K\I,for which suppbj consists entirely of boundary and exterior cells.Weighted Functions A weight functionωof order y∈No is continuous on D and satisfiesω(x) (?) dist(x,Γ)y, x∈DWeighted Extended B-Splines For an outer index j∈J, let I(j)= l+{0.…, n}m (?) I be an m-dimensional array of inner indices closest to j,assuming that h is small enough so that such an array exists. Moreover, denoted by the values of the Lagrange polynomials associated with I(j) and by J(i) the set of all j with i∈l(j). Then, the web-splines form a basis for the web-spaceωeBhn(D). Actually,hereω(xi)≡1 from the definition of our weight function. Construct Weighted Function of C1 A SplinesThis paper constructs C1 algebra splines as weighted function on the boundary of the target region.,where the boundary surface has thickness,consist of 3-prisms and 4-prisms. The direction of the edge is coincident with the given normal of the point at which the edge and the boundary intersect.Moreover,each edge extends the same length into and outward the boundary surface.Finally,we get such a function F: (1) F is defined piecewise on the prisms,totally C1; (2)When F(x,y,z)=±1,▽F= 0; (3)F(x,y,z)=0 interpolates the given points and the given normals. Fig 1Coordinate transformation In order to facilitate construction,we firstly define coordinate system. egde cellsVi={p:p=Qi(λ)=Qi+λsiNi,λ∈[-1,1]} face cellsHij={p:p=(1-t)Qi(λ)+tQj(λ),t∈[0,1],λ∈[-1,1]} On the volume cells,we define functions respectively on the 3-primes and 4-prisms. 3-prims Dijk: Dijk={p:p=biQi(λ)+bjQj(λ)+bkQk(λ),bi,bj,bk∈[0,1],bi+bj+bk=1,λ∈[-1,1]} 4-prisms Dijkl: Dijkl={p:p=B00(u,v)Qi(λ)+B10(u,v)Qj(λ) +B01(u,v)Ql(λ)+B11(u,v)Qk(λ), u,v∈[0,1],λ∈[0,1]}Then,we construct target functions on these prims.The construction is stepwise.First,the function is defined on the edges of the volumes,the on the faces and finally in the vol-umes.On the edge cells On the edge cell Q(λ),we get F(Q(λ))by Hermit interpolation of single variable. Here Hi3(λ) are the basis functions of cubic Hermit interpolation on the interval [-1,1]:On the face cells On the face cell Hij,we get the function Fij by interpolating Q(λ) and Q(λ): Here Hi3(t) are the basis functions of cubic Hermit interpolation on the interval [0,1]: moreoverOn the prism cells On the 3-prism cell D123,we get F123 by edge-face interpolation; On the 4-prism cell D1234 we get F1234by face-face interpolation On D123: where here we get Di(b1,b2,b3,λ)by interpolating Qi(λ)and the function value of point and the directional derivatives in the direction inequal to each other。that is where here"On D1234: where du(v,λ)=H23(v,λ)-H14(v,λ),dv(u,λ)=H43(u,λ)-H12(u,λ)。Finally define the function in D1234 is: where wu= [(1-v)v]2, wv= [(1-u)u]2 We define and the concepts of "inside" and "outside" are illustrated by Fig2。Fig 2This paper proves in details the satisfaction of the constructed function to the assumes,getting the expected weighted functionω.By adjusting the extending length s,,we can control the extending length of the thick surface to be not larger than h/4.We need to note that,the Her-mit interpolation used in the construction is not exactly the traditional Hermit interpolation.It is a improvement for keeping the continuity on the boundary of the thick surface. This makes the degree of the function higher,but guarantee the C1 continuity. We then get some stability and approximation results of the WEB method under this weighted function. Stability and the Approximation orderDual Functions For any m-dimensional cube Q'i (?) supp(Bi) with width h, there exists a functionλi with support in Q'i which satisfies and‖λi‖0≤const(m,n)h-m/2.Weighted Dual Function For web-splines corresponding to our A-spline as the weight func-tionω, there exist locally supported,uniformly bounded dual functionsΛi,i.e., and suppΛi (?) Qi,‖Ai‖0≤const(D,ω,n)h-m/2.Estimate about condition number As an application of theorem 3.3 and 3.4, we derive a bound for the condition number of the Ritz-Galerkin matrix for web-splines. Since Gh is symmetric, condGh is the quotient of the largest and small-est eigenvalue. These two extreme eigenvalue can be computed by maximizing and minimizing the Rayleigh quotient The numerator is≤‖vh‖12, so that r(C)≤h-2hm by the theorem 3.4. On the other hand ,since vh vanishes on (?)D, we have by the Poincare-Friedrichs inequality A.12. Hence, r(C)≥hm by the theorem 3.3. This shows thatApproximation order Set For our weight function(A-spline), if v= u/co is smooth on D, then forκ= 0,1. This implies, in particular, that web-splinesωeBh and weighted splinesωBh have the optimal approximation order.The accuracy of weighted spline approximations to a function u depends on the regularity of the quotient v=u/ω.since the division byωcauses a loss of regularity near the boundary even ifωis C1. So the main work in the future is to construct A-splines to obtain v∈Hn+1 or find another way to get the similar optimal order.