High-order space-time finite element schemes for acoustic and viscodynamic wave equations with temporal decoupling

Copyright @ 2014 The Authors. This is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited.We revisit a method originally introduced by Werder et al. (in Comput. Methods Appl. Mech. Engrg., 190:6685–6708, 2001) for temporally discontinuous Galerkin FEMs applied to a parabolic partial differential equation. In that approach, block systems arise because of the coupling of the spatial systems through inner products of the temporal basis functions. If the spatial finite element space is of dimension D and polynomials of degree r are used in time, the block system has dimension (r + 1)D and is usually regarded as being too large when r > 1. Werder et al. found that the space-time coupling matrices are diagonalizable over inline image for r ⩽100, and this means that the time-coupled computations within a time step can actually be decoupled. By using either continuous Galerkin or spectral element methods in space, we apply this DG-in-time methodology, for the first time, to second-order wave equations including elastodynamics with and without Kelvin–Voigt and Maxwell–Zener viscoelasticity. An example set of numerical results is given to demonstrate the favourable effect on error and computational work of the moderately high-order (up to degree 7) temporal and spatio-temporal approximations, and we also touch on an application of this method to an ambitious problem related to the diagnosis of coronary artery disease

Backward Euler method for the Equations of Motion Arising in Oldroyd Fluids of Order One with Nonsmooth Initial Data

In this paper, a backward Euler method is discussed for the equations of motion arising in the 2D Oldroyd model of viscoelastic fluids of order one with the forcing term independent of time or in $L^{\infty}$ in time. It is shown that the estimates of the discrete solution in Dirichlet norm is bounded uniformly in time. Optimal a priori error estimate in L2-norm is derived for the discrete problem with non-smooth initial data. This estimate is shown to be uniform in time, under the assumption of uniqueness condition

Hyperbolic Balance Laws: modeling, analysis, and numerics (hybrid meeting)

This workshop brought together leading experts, as well as the most promising young researchers, working on nonlinear hyperbolic balance laws. The meeting focused on addressing new cutting-edge research in modeling, analysis, and numerics. Particular topics included ill-/well-posedness, randomness and multiscale modeling, flows in a moving domain, free boundary problems, games and control

On the numerical approximation of a problem involving a mixture of a MGT viscous material and an elastic solid

The version of record os available online at: https://doi.org/10.1007/s40314-022-01784-8In this work, we analyze, from the numerical point of view, a problem including a mixture made of a MGT viscoelastic solid and an elastic solid. The corresponding variational problem is a linear system composed of two coupled hyperbolic equations written in terms of the acceleration of the first constituent and the velocity of the second one. Then, fully discrete approximations are introduced by using the finite element method and the implicit Euler scheme. A discrete stability property and a priori error estimates are proved. Finally, some one-dimensional numerical simulations are shown to demonstrate the accuracy of the proposed approximations and the behaviour of the solutionPeer ReviewedPostprint (author's final draft