1 136 0.0011424
2 264 0.0004612
3 403 0.0000822
4 555 0.0000248
Table 4.5: H0-error for adaptive method over strongly nested domains withh0= 1/6.
Hierarchical levels DOF(Adaptive Method) H0-error
1 136 0.0011424
2 427 0.0007591
3 1563 8.2702 e -6
Table 4.6: H0-error for adaptive method withh+= 1/6 andtol.= 10−2.
Table4.6shows the DOF and theH0-error for the solution of the Poisson problem solved over the nested domain obtained by comparing the approximationuhwith the exact solutionu. The nested domains have the grid widthsh+= 1/6 andh− = 1/24.
As for the previous example, the tables confirm a slightly better performance of our hierarchical approximations compared to the WEB method.
44 CHAPTER 4. IMPLEMENTATION
Chapter 5
Summary and Discussion
This chapter outlines the contents of the thesis which discussed the finite element approximation of an elliptic boundary value problem by using hierarchical weighted B-splines. It was often asked about the adaptivity of WEB-splines for boundary value problems. We developed a scheme of adaptivity for weighted linear B-splines.
In the second chapter, we gave a short introduction to the theory of the finite element method.
As a model problem, we considered Poisson’s equation with Dirichlet boundary conditions
−∆u=f in D u= 0 on ∂D on a bounded domainD⊂R2.
Since, in general, there does not exist a classical solution of this boundary value problem, it is solved in a generalized function space, the Sobolev space, which is discussed in section 2.2.
Its definition uses a weak derivative, a generalization of the classical derivative via Lebesgue integration by parts.
The variational formulation of our model problem is Z
D∇u.∇v dD= Z
D
f v dD ∀v∈H01(D).
The Lax-Milgram theorem guarantees the existence of a unique solution u of the variational problem
J(u) =minv∈VJ(v), J(v) = 1
2a(v, v)−F(v) 45
46 CHAPTER 5. SUMMARY AND DISCUSSION and hence of our model problem.
For the computation of an approximate solutionuh, the Ritz-Galerkin method employs a finite subspace Vh of H01(D) using basis functions bk with the local support. Then uh has a unique representation of the formuh=PNh
k=1ckbk, whereNh=dim(Vh). Substitutinguhin variational formulation and restrictingv to{b1,· · · , bNh}, one obtains the Galerkin system
Gu=f
where the entries ofG are given by (4.16). Lemma2.4.1 shows the positive definiteness of the stiffness matrix G. According to Cea’s lemma2.4.2, the error of the Ritz-Galerkin solution uh
corresponds to the error of the best approximation ofuinVh.
B-splines play an important role. We use B-splines as finite element basis functions. Two major problems arise when using B-splines as basis functions in finite element approximations on arbitrary domains. One is the fulfillment of homogeneous Dirichlet boundary conditions and the other is the stability problem.
We can remove these difficulties by simple modifications. The homogeneous Dirichlet bound-ary condition causes, in general, all the coefficients near the boundbound-ary to be zero. As a result, the approximation power is lost. This deficit can easily be removed by multiplying the B-splines with a sufficiently smooth weight function which vanishes on the boundary and is positive inside D. For domains with smooth boundary∂D, the distance function can be used as a weight func-tion. An alternative approach for generating weight functions is the R-function method. This technique is useful for the domains designed by CSG techniques. In this case, the R-function method supplies set operations to construct a global weight function from the boundary segments.
Weighting the B-splines leads us to the weighted B-spline basis (WB-basis), which incorporates essential boundary conditions to the finite element basis. The Bramble-Hilbert’s lemma helps us to find the error bound foru∈Hn(D) using the orthogonal projectionLnuonto polynomial Pn−1(D) of total degree < n on D. The error bound for the Sobolev norm on Hℓ(D) for 0≤ℓ < k≤nis
ku−LnukHℓ(D)≤const(D, n)̺k−ℓkukHk(D) for 0≤ℓ < k≤n,
47 where ̺is the diameter of the domainD. For the spline function uh =P
k∈Kckbk,h, the error bound foru∈H2(D) is
ku−X
k∈K
ckbk,hkHℓ(D)h2−ℓkukH2(D) f or 0≤ℓ <2.
In section 3.5, we explain the adaptive refinement with hierarchical bases. For the refinement step some of the B-splines on the coarser grid are subdivided. The subdivision is accomplished by the formula
bk,h(x) = X2 l=0
slb2k+l,h/2(x) where
sl= Y2 ν=1
1 4
2 lν
.
If byUh we denote the coefficients of a spline functionuh=P
kuk,hbl,h on the coarser grid, thenuhis represented on the fine grid by the representation (5.6).
The coefficients on the fine grid and on the coarse grid are related by Uh/2=P Uh, pk,l=sk−2l.
In adaptive finite element computations, we adjust the spatial resolution of the domain dis-cretization to obtain high accuracy on some subregions ofD. To fulfill this requirement we insert a new set of basis functions having a smaller grid width than the coarser basis functions.
We define hierarchical splines in definition3.5.1corresponding to a nested sequence of domains D:Dh+⊃Dh+/2⊃...⊃Dh−.
We replace the relevant B-splinesbk,h withk ∈Kh by the finer B-splines via subdivision. The spanning set is then
bk,h, k∈Kh, h∈ {h+h+/2,· · · , h−}.
The hierarchical B-splines which spanB(h+,h−)(D) are linearly independent. We prove the linear independence in theorem3.5.3.
To keep the basis uniformly conditioned, we impose some restrictions on the nested sequence of domains. We introduced strongly nested domains in definition3.6.1.
Strongly nested domains obey the following properties:
48 CHAPTER 5. SUMMARY AND DISCUSSION
• B-splines in the basis with non-empty intersection of their supports must belong to two consecutive hierarchical levels.
• The distance between the boundary∂Dh and∂Dh/2 is greater than 2h.
The error of a hierarchical approximationpto a smooth bivariate functionf is bounded by
|f(x)−p(x)| ≤const h2 X2 ν=1
k∂ν2fek∞,Dx,h, x∈Dh\Dh/2.
A proof of this statement is given in theorem 3.6.1.
Chapter 4 is devoted to the numerical implementation of an adaptive scheme. We used MAT-LAB as a software package. In section 4.1 we explain the data structure to store the information obtained from the adaptive grid discretization and the resulting spline space. The grid data consists of an array of structuresHB(ℓ), where ℓdenotes the discretization level. To assemble the Ritz-Galerkin system we make a list of all relevant B-splines in the form of triplets (k, ν, µ) containing the levelk and the index (ν, µ) of the B-spline. The entries of the Galerkin system
gk,l= Z
D
gradbkgradbl
are computed with the help of numerical integration. The techniques for numerical integration over grid cells is explained in section 4.3. The Galerkin matrix is obtained by adding the con-tributions from each grid cell. We conclude by describing our refinement strategy and some numerical examples.