Function SpacesΒΆ

To use the boundary element method, we start with a variational boundary integral equation, for example: Find \(u\in H^{1/2}(\Gamma)\) such that for all \(v\in H^{1/2}(\Gamma)\),

\[ \left\langle\mathsf{V}u,v\right\rangle = \left\langle f,v\right\rangle. \]

An approximation of the solution of this problem is then found by discretising the problem and searching for a solution in a subspace \(\mathcal{V}_h\subset H^{1/2}(\Gamma)\).

In this section of the Bempp Handbook, we look at the definitions of continuous function spaces such as \(H^{1/2}(\Gamma)\) that are used in the boundary integral equations, and we look at how discrete subspaces of these are usually defined.