Adaptive Galerkin Finite Element Methods for the Wave Equation

Abstract

-This paper gives an overview of adaptive discretization methods for linear second-order hyperbolic problems such as the acoustic or the elastic wave equation. The emphasis is on Galerkin-type methods for spatial as well as temporal discretization, which also include variants of the Crank-Nicolson and the Newmark finite difference schemes. The adaptive choice of space and time meshes follows the principle of "goaloriented" adaptivity which is based on a posteriori error estimation employing the solutions of auxiliary dual problems.2000 Mathematics Subject Classification: 35L05, 65M50, 65M60, 74S05.

Error Assessment in Structural Transient Dynamics

“…For instance, the Newmark method is not using any time variational form. However, there are some a posteriori error estimation techniques using the full variational formulation both for the problem approximation and for the error assessment strategy, see [35,50,29,30].…”

“…The other three options follow a Double Field (DF) formulation. The second option, based on [35], is a time-Continuous Galerkin approach using the mass product m(·, ·) to enforce the displacement-velocity consistency, and it will be referred as CGM. The third option, denoted by CGA, very similar to the previous one, differs in the fact that displacement-velocity consistency is enforced using the bilinear form a(·, ·), see [50].…”

“…With these notations, the double field time continuous Galerkin weak form of problem (1) presented in [35] reads: find U ∈ W 0 × W 0 such that for all…”

“…Recall that the Newmark method requires a single problem in V H 0 at each time step. However, for q = 1 the block system of algebraic linear equations of double size can be pre-processed to make it equivalent to the Newmark method with parameters β = 1/4 and γ = 1/2, see [35] for a detailed proof.…”

“…In this context, different estimates provide also indicators driving mesh adaptive procedures, either using energy-like measures [29,30,31,32,33,34] or QoI [35,36,37,38]. Estimates providing error bounds are also available both for energy-like error measures [39,40,41] and goal-oriented ones [42,43,44,45,46,47,31].…”