Convergence, Accuracy and Stability of Finite Element Approximations of a Class of Non-linear Hyperbolic Equations

This paper is concerned with the estimation of the numerical stability and rate-of-convergence of finite element approximations of transient solutions of a rather wide class of non-linear hyperbolic partial differential equations. The equations of motion of practically all homogeneous hyperelastic bodies, including both physically non-linear materials and finite amplitude motions, fall within the general class of equations considered here; but, for simplicity, we limit ourselves to one-dimensional bodies (i.e. one-dimensional spatial domains) and we assume that the initial data and the solution are smooth functions of x and t. Thus, while we rule out sharp discontinuities such as shocks, our results are perfectly valid for most transient non-linear vibration problems. The essential feature of the present analysis is that full discretization in both space (i.e. in the particle labels x) and in time t is used: the finite element method is used to approximate the variation of the solution in x, while ordinary central differences are used to approximate various time rates-of-change. The study of accuracy and convergence of finite element approximations has, until recent times, been largely confined to linear strongly elliptic operators (see, for example References 1-9). Generalizations of certain results to classes of non-linear elliptic-type problems have been discussed by Ciarlet and coworkers,1°-13 Melkes,H Oden15.16 and Varga,1? but extensions to problems of the evolution type have come about much more slowly. The work of Douglas and Dupont18 provides a basis for deriving error estimates for linear and certain non-linear parabolic equations, and Kikuchi and Ando19 presented a penetrating study of proper lies of finite element approximations of a class of linear and non-linear equations of evolution. Fix and Nassif investigated finite element approximations of certain linear parabolic equations20 and linear first-order hyperbolic equations.21 More recently, Fujii22 examined the stability and convergence of finite element approximations of smooth solutions of linear second-order hyperbolic equations in which Newmark's ,B-method is used to represent the behaviour in time. Like Fujii, we also examine stability in certain natural energy norms; our error estimates essentially agree with those Fujii obtained for the lincar case, but our approuch is necessarily quite different. We confine our attention to a class of non-linear wave equations of a form very common in continuum mechanics, in which the wave speed is a bounded, continuous, and always positive function of the gradient liz of the dependent variable lI(x, t), and for which u(x, t) has continuous third derivatives with respect to time t. The temporal behaviour of lI(x, t) is approximated using standard central differences and the spatial