Mixed finite element solution of time-dependent problems

A mixed formulation of the finite element method is used to establish a higher-order incremental method for the solution of second-order/hyperbolic problems. The displacement, the velocity and, optionally, the acceleration fields are approximated independently in time using hierarchical bases. The time approximation criterion preserves hyperbolicity in the sense that it replaces the solution of hyperbolic problems by the solution of uncoupled Helmholtz-type elliptic problems, which can be subsequently solved using the alternative methods currently in use for discretization of the space dimension. The development of the time integration procedure, the characterization of its performance in terms of stability, accuracy and convergence are illustrated using a polynomial time basis. In order to stress the fact that the procedure can be implemented using alternative time bases, a wavelet system is used in the solution of nonlinear, parabolic and hyperbolic problems. The method is well-suited to parallel processing and to large time stepping. The extension of its application to the solution of other than linear second-order/hyperbolic problems is discussed.