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# Finite Element Methods with Continuous Displacement, P

DOI link for Finite Element Methods with Continuous Displacement, P

Finite Element Methods with Continuous Displacement, P book

# Finite Element Methods with Continuous Displacement, P

DOI link for Finite Element Methods with Continuous Displacement, P

Finite Element Methods with Continuous Displacement, P book

## ABSTRACT

P. Joly, POEMS Project team, INRIA-Rocquencourt, France, [email protected]

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,

L(t) ∈ C1(IR+;V ′)

(9.32) holds with ν = 0, i.e., a is coercive. (10.1)

Principle of the method. This method is linked to the existence of {Vh, 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 Vh approaches V as h goes to 0.