Finite element methods with continuous displacement

We develop in this chapter the very well-known basic principles of the standard Lagrange finite element approximation of elastodynamics equations in their form (9.24), i.e., as a second order hyperbolic system in the displacement field. In fact, this method applies to any abstract second order variational evolution problem of the form (9.26) whose a particular case is the variational (or weak) formulation of the second order elastodynamics system (9.24). That is why we choose a general abstract presentation in the next paragraphs, assuming that the general assumptions (9.31) to (9.32) hold. Of course, the space V is supposed to be of infinite dimension, as it is the case of the space V defined by (9.25). For the simplicity of the exposition, we shall use some stronger hypotheses that are needed (see theorem 9.1), namely, (equation found) (9.32) holds with ν = 0, i.e., a is coercive. Principle of the method. This method is linked to the existence of {V_h, h > 0} finite dimensional subspaces of V where h is an approximation parameter devoted to tend to zero. From the theoretical point of view, it suffices that h describes a subset of IR⁺ having 0 as accumulation point. In practice, for finite element methods, h will be the stepsize of a spatial mesh of the computational domain Ω. The important property from the theoretical point of view is a consistency condition meaning that V_h approaches V as h goes to 0.