The Standard Galerkin Method

In this introductory chapter we shall study the standard Galerkin finite element method for the approximate solution of the model initial-boundary value problem for the heat equation, (1.1) $$\eqalign{ & u_t - \Delta u = f{\text{ }}in\:{\text{ }}\Omega ,\:{\text{ }}for\:t > 0, \cr & u = 0\:on\:\partial \Omega ,\:for\:t > 0,\:with\:u(\cdot,0) = v\:in\:\Omega \cr} $$ where is a domain in Rd with smooth boundary ∂Ω, and where u = u(x, t), u _t denotes ∂u/∂t, and $$ \Delta = \sum\nolimits_{j = 1}^d {\partial ^2 /\partial x_j^2 } $$ the Laplacian.