in two steps. First, we will show that the Galerkin equation is a well-posed problem in the sense of Hadamard and therefore admits a unique solution. In the second step, we study the quality of approximation of the Galerkin solution . The analysis will mostly rest on two properties of the bilinear form, namely Boundedness: for all holds for some constant Ellipticity: for all holds for some constant By the Lax-Milgram theorem (see weak formulation), these two conditions imply well-posedness of the original problem in weak formulation. All norms in the following sections will be norms for which the above inequalities hold (these norms are often called an energy norm). Well-posedness of the Galerkin equation Since , boundedness and ellipticity of the bilinear form apply to . Therefore, the well-posedness of the Galerkin problem is actually inherited from the well-posedness of the original problem. Quasi-best approximation (Céa's lemma) The error between the original and the Galerkin solution admits the estimate This means, that up to the constant , the Galerkin solution is as close to the original solution as any other vector in . In particular, it will be sufficient to study approximation by spaces , completely forgetting about the equation being solved. Proof Since the proof is very simple and the basic principle behind all Galerkin methods, we include it here: by ellipticity and boundedness of the bilinear form (inequalities) and Galerkin orthogonality (equals sign in the middle), we have for arbitrary : Dividing by and taking the infimum over all possible yields the lemma. Galerkin's best approximation property in the energy norm For simplicity of presentation in the section above we have assumed that the bilinear form is symmetric and positive definite, which implies that it is a scalar product and the expression is actually a valid vector norm, called the energy norm. Under these assumptions one can easily prove in addition Galerkin's best approximation property in the energy norm. Using Galerkin a-orthogonality and the Cauchy–Schwarz inequality for the energy norm, we obtain Dividing by and taking the infimum over all possible proves that the Galerkin approximation is the best approximation in the energy norm within the subspace , i.e. is nothing but the orthogonal, with respect to the scalar product , projection of the solution to the subspace . Galerkin method for stepped Structures I. Elishakof, M. Amato, A. Marzani, P.A. Arvan, and J.N. Reddy studied the