12.4 Galerkin’s method and finite elements
The weak form suggests a numerical method. Choose a finite-dimensional subspace \(V_h \subset H^1_0\) and find \(u_h \in V_h\) with \[ (u_h, v)_E = \ell (v) \qquad \text {for all } v \in V_h . \] With a basis \(\phi _1, \ldots , \phi _N\) of \(V_h\) and \(u_h = \sum U_j\phi _j\), this is the linear system \(KU = b\) with stiffness matrix \(K_{ij} = (\phi _i, \phi _j)_E\) and load vector \(b_i = \ell (\phi _i)\). \(K\) is symmetric and positive definite, so the system has exactly one solution.
Theorem 12.4.1 (Céa). The Galerkin solution is the best approximation to \(u\) from \(V_h\) in the energy norm: \(\|u - u_h\|_E = \min _{v \in V_h}\|u - v\|_E\).
Proof. Subtracting the equations for \(u\) and \(u_h\) gives Galerkin orthogonality, \((u - u_h, v)_E = 0\) for all \(v \in V_h\). So \(u_h\) is the orthogonal projection of \(u\) onto \(V_h\) in the energy inner product, and orthogonal projection gives the nearest point. □
In the finite element method, \(V_h\) consists of continuous piecewise polynomials on a mesh. In one dimension with nodes \(x_i = ih\), the piecewise linear hat functions \(\phi _i\), equal to \(1\) at \(x_i\) and \(0\) at the other nodes, give \(K = \frac 1h\operatorname {tridiag}(-1, 2, -1)\) (Figure 15). Since the interpolant of a smooth \(u\) is within \(O(h)\) of it in the energy norm, Céa’s theorem gives \(\|u - u_h\|_E = O(h)\), and a duality argument (Aubin and Nitsche) gives \(\|u - u_h\|_{L^2} = O(h^2)\). In two and three dimensions the mesh is of triangles or tetrahedra fitted to the region, which is what makes the method so flexible.
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