9 research outputs found
Parallel space-time solutions for the linear visco-acoustic and visco-elastic wave equation
We present parallel adaptive results for a discontinuous Galerkin space-time discretization for acoustic and elastic waves with attenuation. The method is based on -adaptive polynomial discontinuous ansatz and test spaces and a first-order formulation with full upwind fluxes. Adaptivity is controlled by dual-primal error estimation, and the full linear system is solved by a Krylov method with space-time multilevel preconditioning. The discretization and solution method is introduced in Dörfler-Findeisen-Wieners (Comput. Meth. Appl. Math. 2016) for general linear hyperbolic systems and applied to acoustic and elastic waves in Dörfler-Findeisen-Wieners-Ziegler (Radon Series Comp. Appl. Math. 2019); attenuation effects were included in Ziegler (PhD thesis 2019, Karlsruhe Institute of Technology). Here, we consider the evaluation of this method for a benchmark configuration in geophysics, where the convergence is tested with respect to seismograms. We consider the scaling on parallel machines and we show that the adaptive method based on goal-oriented error estimation is able to reduce the computational effort substantially
A cVEM-DG space-time method for the dissipative wave equation
A novel space-time discretization for the (linear) scalar-valued dissipative wave equation is presented. It is a structured approach, namely, the discretization space is obtained tensorizing the Virtual Element (VE) discretization in space with the Discontinuous Galerkin (DG) method in time. As such, it combines the advantages of both the VE and the DG methods. The proposed scheme is implicit and it is proved to be unconditionally stable and accurate in space and time
Space-time discontinuous Petrov-Galerkin methods for linearwave equations in heterogeneous media
We establish an abstract space-time DPG framework for the approximation of linear waves in heterogeneous media. The estimates are based on a suitable variational setting in the energy space. The analysis combines the approaches for acoustic waves in Gopalakrishnan / Sepulveda (A space-time DPG method for acoustic waves, arXiv 2017) and in Ernesti / Wieners (RICCAM proceedings, submitted 2017) and is based on the abstract definition of traces on the skeleton of the time-space sub-structuring. The method is evaluated by large-scale parallel computations motivated from applications in seismic imaging, where the computational domain can be restricted substantially to a subset of the full space-time cylinder
A space-time quasi-Trefftz DG method for the wave equation with piecewise-smooth coefficients
Trefftz methods are high-order Galerkin schemes in which all discrete
functions are elementwise solution of the PDE to be approximated. They are
viable only when the PDE is linear and its coefficients are piecewise constant.
We introduce a 'quasi-Trefftz' discontinuous Galerkin method for the
discretisation of the acoustic wave equation with piecewise-smooth wavespeed:
the discrete functions are elementwise approximate PDE solutions. We show that
the new discretisation enjoys the same excellent approximation properties as
the classical Trefftz one, and prove stability and high-order convergence of
the DG scheme. We introduce polynomial basis functions for the new discrete
spaces and describe a simple algorithm to compute them. The technique we
propose is inspired by the generalised plane waves previously developed for
time-harmonic problems with variable coefficients; it turns out that in the
case of the time-domain wave equation under consideration the quasi-Trefftz
approach allows for polynomial basis functions.Comment: 25 pages, 9 figure
Space-time discontinuous Galerkin approximation of acoustic waves with point singularities
We develop a convergence theory of space-time discretizations for the linear,
2nd-order wave equation in polygonal domains ,
possibly occupied by piecewise homogeneous media with different propagation
speeds. Building on an unconditionally stable space-time DG formulation
developed in [Moiola, Perugia 2018], we (a) prove optimal convergence rates for
the space-time scheme with local isotropic corner mesh refinement on the
spatial domain, and (b) demonstrate numerically optimal convergence rates of a
suitable \emph{sparse} space-time version of the DG scheme. The latter scheme
is based on the so-called \emph{combination formula}, in conjunction with a
family of anisotropic space-time DG-discretizations. It results in
optimal-order convergent schemes, also in domains with corners, with a number
of degrees of freedom that scales essentially like the DG solution of one
stationary elliptic problem in on the finest spatial grid. Numerical
experiments for both smooth and singular solutions support convergence rate
optimality on spatially refined meshes of the full and sparse space-time DG
schemes.Comment: 38 pages, 8 figure
Space-time discontinuous Galerkin approximation of acoustic waves with point singularities
We develop a convergence theory of space-time discretizations for the linear,
2nd-order wave equation in polygonal domains ,
possibly occupied by piecewise homogeneous media with different propagation
speeds. Building on an unconditionally stable space-time DG formulation
developed in [Moiola, Perugia 2018], we (a) prove optimal convergence rates for
the space-time scheme with local isotropic corner mesh refinement on the
spatial domain, and (b) demonstrate numerically optimal convergence rates of a
suitable \emph{sparse} space-time version of the DG scheme. The latter scheme
is based on the so-called \emph{combination formula}, in conjunction with a
family of anisotropic space-time DG-discretizations. It results in
optimal-order convergent schemes, also in domains with corners, with a number
of degrees of freedom that scales essentially like the DG solution of one
stationary elliptic problem in on the finest spatial grid. Numerical
experiments for both smooth and singular solutions support convergence rate
optimality on spatially refined meshes of the full and sparse space-time DG
schemes.Comment: 38 pages, 8 figure