This section establishes a connection between the Laplacian operator ∆ and the Laplacian matrix4 Lby techniques from engineering analysis calledfinite element methods, see, e.g., [21, 27]. On the one hand, this explains why and how we can use L to approximate ∆ in PDEs, on the other hand, we also obtain a measure on the quality of a finite approximation, which is given by a bound on the difference to the exact solution with respect to some norm. We illustrate this for the potential equation. The wave equation can be handled similarly. We sketch a method that is a special case of the Ritz-Galerkin method. Assume that a classical Dirichlet problem (Ω, g) with a bounded domain Ω has a solutionf ∈ F :=C2(Ω)∩ C(Ω) that minimizes the Dirichlet integral (which in most cases holds unless we artificially construct a counterexample, see Sect. 1.4 on Dirichlet’s principle). Then
f = arg min
h∈F h|∂Ω=g
D(h) = arg min
h∈F h|∂Ω=g
Z
x∈Ω
(∇h(x))T∇h(x) dx . (2.28)
We consider the case Ω ⊆ R2. For n 6= 2 the method basically carries over easily, but the choice of the primitives, the finite elements, has to be adapted.
Assume that Ω is well approximated by (closed) equal sized squares with side length δ > 0, such that ideally Ω is the union of these squares, and the cut of any two distinct squares is either empty, a corner, or an edge of a square.
We assume that Ω is exactly the union of these squares (otherwise, let some squares at the boundary ∂Ω be slightly deformed and we may just replace the symbol = by≈for some of the following integrals). Split each square into two trianglesTj, see also Fig.2.10. In our case, where finiteness is trivial, and in the general case, there are only finitely many finite elements Tj. If we had only some given finite point setV on Ω⊆R2, e.g., aDelaunay triangulation may be used to obtain finite elements. Now define the finite dimensional
4Within finite element methods the Laplacian matrix is calledstiffness matrix.
Figure 2.10: Finite element method for the Dirichlet problem
function space Fδ by
Fδ :={h∈ C(Ω) : ∇h|T◦j is constant for allj}.
For the following, Fδ must be chosen as a finitely generated subset of F, which is not the case here because functions inFδ are in general not inC2(Ω).
We fix this by F := W1,2(Ω)∩ C(Ω). We need this space only for technical purposes in the following lemmata. (Otherwise, for the moment, think of keeping the original space F and smoothening the functions in Fδ at ∂Tj.) Define the graphGδ = (V, E) with vertices v1, . . . , vn that are thecorners of all Tj (where corners at the same position are identified) and with edges E induced by the edges of all squares. For each triangle Tj let v0,j be the vertex at the right angle, and let v1,j, v2,j be its neighbors in Tj. Now we can reformulate the minimum from (2.28), where we identify vertices with their positions, and h∈ Fδ with the corresponding function h onV.
f = min
Note that each edge that is counted only once instead of twice (when summing over all triangles) is on ∂Ω, and the values of its end vertices are fixed, such that the minimum is not affected. The area δ2/2 of the triangles takes care that we do not need a correction factor between −∆ and L for n = 2.
Anyhow, when we apply this technique withn-dimensional hypercubes inRn, we need a factor ofδ2−n. Finally, we state two of the most prominent lemmata used for finite element methods. Proofs are found in, e.g., [27].
Lemma 2.7.1 (Lax-Milgram) Let H be a Hilbert space with norm | · |H.
How does this apply to our case? Let H := W01,2(Ω) denote the closure ofC0∞(Ω) with respect to theW1,2-norm (functions fromW1,2(Ω) that vanish to the boundary ∂Ω), and let| · |H:=| · |W1,2. Moreover, letf ∈ C2(Ω)∩ C(Ω) be the solution of the classical Dirichlet problem (Ω, g) given above. Note that f|Ω∈ W1,2(Ω). Furthermore,
Clearly,a is bilinear and ` is linear and continuous. By the Cauchy-Schwarz inequality, continuity ofawithc1 = 1 is also straight-forward. To show thata is also coercive we need information about Ω and the following inequality.
Theorem 2.7.2 (Poincar´e-Friedrichs) For allh∈ W01,2(Ω)there exists a constant C > 0, such that
Z
If we write the inequality in terms of inner products and furthermore assume that 06=h∈ W02,2(Ω), we recognize the relation to eigenvalues of −∆.
(h,−∆h)L2(Ω) (h, h)L2(Ω)
≥ 1 C . Now we can show that a is coercive.
|h|H ≤ |h|L2(Ω)+ In general, it is not an easy task to determine C. Roughly speaking, C can be chosen as the inverse of the smallest eigenvalue of −∆. For our example from the last section, Ω = (0, a)×(0, b), we can useC = ((π/a)2+ (π/b)2)−1
and thus define c2 := 1/(C+ 1)2. A lower bound for the smallest eigenvalue ofplane-covering domains (e.g., triangles, rectangles, regular hexagons, etc.), is given, e.g., in [101], by 4π/|Ω|, where |Ω| is the area of Ω.
Together with the second lemma, this gives a bound on the quality of an approximation fδ ∈ Fδ (see (2.29)).
Lemma 2.7.3 (C´ea) With the assumptions of Lemma 2.7.1 let u ∈ H be the unique solution of a(u, h) = `(h) for all h∈ H. Furthermore, for a finite dimensional subspace Hδ⊆ H let a(uδ, hδ) = `(hδ) for all hδ ∈ Hδ. Then
|u−uδ|H ≤ c1 c2 · inf
hδ∈Hδ|u−hδ|H . Furthermore assume that
limδ→0 inf
hδ∈Hδ|h−hδ|H= 0 for all h∈ H ,
which holds in our case (if it is possible to fill the gap between boundary ∂Ω and interior with primitives of a diameter bounded by δ). Extendg ∈ C(∂Ω) arbitrarily to W1,2(Ω)∩ C(Ω). Then foru=f−g ∈ H andfδ:=uδ+g ∈ Fδ
(now Fδ := {hδ+g : hδ ∈ H}) Lemma 2.7.3 implies limδ→0|f −fδ|H = 0, which means that the finite case solutions fδconverge to the continuous case solution f with a quality bound of |f−fδ|H≤(c1/c2) infhδ∈Fδ|f −hδ|H.
Note that it is often possible to bound infhδ∈Fδ|f−hδ|Hwithout knowing the solution f. We sketch this for the example given in the proof of Exam-ple 1.4.2, and depicted in Fig. 2.10, where Ω = B(0, ε,1) ⊆ R2,0 < ε < 1, and g = 0 on the outer circle, g = 1 on the inner circle. First we transform to polar coordinates (r, ϕ). By symmetry to 0 and the maximum princi-ple, f(r,·) is constant. Again by the maximum principle and given boundary conditions, f(·, ϕ) is strictly decreasing and bounded into [0,1]. LetGδ, for simplicity, be a grid graph in polar coordinates, i.e., in Cartesian coordinates, set the vertices ofGδradially to0, equidistantly on every circle with center0, and equidistantly on every radius. Let edge lengths be bounded by δ≤1/2, and let the straight-line embedding be maximally contained in Ω. Then, we can easily give a rough lower bound of the infimum by 2πδ. Finally, we can also give a constantc2 to apply the lemma of C´ea. The eigenvalues and eigen-functions of−∆ wrt the domain Ω can be given by means of Bessel functions Jk of the first kind, see, e.g., [28] for details. Assume for simplicity that ε is equal to the quotient q0/q1 ≈ 0.436 of the first two roots q0 ≈ 2.405 and q1 ≈5.520 of J0. Then we can define c2 = 1/(q1+ 1)2, such that altogether
|f −fδ|H ≤(q1+ 1)2·2πδ .