A weakly inhomogeneous vibrating membrane and the solotone effect in two dimensions

A The Galerkin method

Throughout this paper we consider a special case of the eigenvalue problem

defined over a region R. Here u is a function of three spatial variables x, y and z. We shall assume that u = 0 on ∂ ⁡ R 1 , that is on part of the boundary of R. On the remaining part of the boundary, ∂ ⁡ R 2 , we impose the condition D ⁢ ( x , y , z ) ⁢ ∂ ⁡ u ∂ ⁡ n = - p ⁢ ( x , y , z ) ⁢ u , or alternatively, D ⁢ ∇ ⁡ u ⋅ 𝐧 = - p ⁢ u . Our aim is to find the eigenvalues of equation (A.1), denoted by λ. This example is not new, see [20, Section 5.10], but our analysis of the resulting eigenspectra is novel.

To derive the weak formulation of equation (A.1) we multiply the partial differential equation and second boundary condition by a smooth arbitrary function Φ, which we assume is zero on ∂ ⁡ R 1 . The resulting equation is then integrated over R to give, after some manipulation,

Equation (A.2) is called the weak formulation of equation (A.1).

The Galerkin method attempts to approximate the eigenvectors and eigenvalues of equation (A.1) by replacing u in equation (A.2) with the weighted sum

Thus

where each Φ i vanishes on ∂ ⁡ R 1 and the functions { Φ 1 , Φ 2 , … , Φ M } form a linearly independent set. Each a i is a scalar quantity to be determined. The function U is only required to satisfy (A.2) for Φ = Φ 1 , Φ 2 , … , Φ M . Therefore, after some manipulation, the weak formulation becomes

We can write this system of M equations more compactly as

where

and

Alternatively, using matrices, 𝐀𝐚 = λ ⁢ 𝐁𝐚 , where 𝐀 = [ A k ⁢ i ] and 𝐁 = [ B k ⁢ i ] are M × M matrices. This generalised matrix eigenvalue problem may be solved using, for example, the inverse power method [20, Section 4.11]. Note, however, this method assumes matrices 𝐀 and 𝐁 possess certain properties (see [20, Theorem 4.11.1] and the related discussion in [20, Section 5.10]).

We now consider the special case a = 0 , ρ = - 1 and ∂ ⁡ R 2 = 0 . Furthermore, working in two dimensions will be sufficient for our purposes. Thus,

Furthermore, equation (A.1) becomes

This is the partial differential equation discussed in Section 3.

B A coarse grid example

The finite element method we will now be illustrated by estimating the smallest eigenvalue of equation (A.1). To achieve this we define R to be the rectangular region shown in Figure 6 but with dimensions a = 3 and b = 1 + 2 . This region will be divided using a finite element grid consisting of five nodes and sixteen triangles, as shown in Figure 16.

The smallest eigenvalue will be approximated using equations (A.5), (A.8) and (A.9). In our case M = 5 , and the matrix elements 𝐀 = [ A k ⁢ i ] and 𝐁 = [ B k ⁢ i ] are to be determined. Each Φ i will be a “pyramid”-like function equal to one at node i and zero at all other nodes. For example, Figure 17 shows Φ 1 (red) and Φ 5 (green).

Figure 17

Functions Φ 1 (red) and Φ 5 (green).

(a)

View from above.

(b)

View from the side.

Performing the necessary integration leads to the following generalised matrix eigenvalue problem:

This problem may be solved using the computer algebra system Maple. In doing so we estimate the smallest eigenvalue of equation (A.1) to be 3.3118. The exact value, calculated using equation (3.5) with D = m = n = 1 , is 2.7900. The relative error is large as one would expect from such a coarse grid. However, our aim was to illustrate the method involved (calculations within the main paper will be performed using a grid consisting of several thousand nodes).

Maple also returns values for a i , i = 1 , 2 , … , 5 . These values may be used to approximate the eigenfunction associated with the smallest eigenvalue (via equation (A.3)). However, our focus remains on the estimation of eigenvalues.