42 research outputs found
A DG-VEM 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
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
A note on optimal spectral bounds for nonoverlapping domain decomposition preconditioners for hp-version discontinuous Galerkin methods
In this article, we consider the derivation of hp-optimal spectral bounds for a class of domain decomposition preconditioners based on the Schwarz framework for discontinuous Galerkin finite element approximations of second-order elliptic partial differential equations. In particular, we improve the bounds derived in our earlier article [P.F. Antonietti and P. Houston, J. Sci. Comput., 46(1):124-149, 2011] in the sense that the resulting bound on the condition number of the preconditioned system is not only explicit with respect to the coarse and fine mesh sizes H and h, respectively, and the fine mesh polynomial degree p, but now also explicit with respect to the polynomial degree q employed for the coarse grid solver. More precisely, we show that the resulting spectral bounds are of order p2H/(qh) for the hp-version of the discontinuous Galerkin method
Dispersion-dissipation analysis of 3D continuous and discontinuous spectral element methods for the elastodynamics equation
In this paper we present a three dimensional dispersion and dissipation analysis for both
the semi discrete and the fully discrete approximation of the elastodynamics equation
based on the plane wave method. For space discretization we compare different approximation
strategies, namely the continuous and the discontinuous spectral element method
on both tetrahedral and hexahedral elements. For time discretization we employ a leapfrog
time integration scheme. Several numerical results are presented and discussed
Discontinuous Galerkin for the heterodimer model of prion dynamics in Parkinson's disease
Neurodegenerative diseases have a significant global impact affecting
millions of individuals worldwide. Some of them, known as proteinopathies, are
characterized by the accumulation and propagation of toxic proteins, known as
prions. Alzheimer's and Parkinson's diseases are relevant of protheinopathies.
Mathematical models of prion dynamics play a crucial role in understanding
disease progression and could be of help to potential interventions. This
article focuses on the heterodimer model: a system of two partial differential
equations that describe the evolution of healthy and misfolded proteins. In
particular, we propose a space discretization based on a Discontinuous Galerkin
method on polygonal/polyhedral grids, which provides flexibility in handling
meshes of complex brain geometries. Concerning the semi-discrete formulation we
prove stability and a-priori error estimates. Next, we adopt a
-method scheme for time discretization. Some convergence tests are
performed to confirm the theoretical bounds and the ability of the method to
approximate travelling wave solutions. The proposed scheme is also tested to
simulate the spread of -synuclein in a realistic test case of
Parkinson's disease in a two-dimensional sagittal brain section geometry
reconstructed from medical images
The conforming virtual element method for polyharmonic and elastodynamics problems: a review
In this paper, we review recent results on the conforming virtual element
approximation of polyharmonic and elastodynamics problems. The structure and
the content of this review is motivated by three paradigmatic examples of
applications: classical and anisotropic Cahn-Hilliard equation and phase field
models for brittle fracture, that are briefly discussed in the first part of
the paper. We present and discuss the mathematical details of the conforming
virtual element approximation of linear polyharmonic problems, the classical
Cahn-Hilliard equation and linear elastodynamics problems.Comment: 30 pages, 7 figures. arXiv admin note: text overlap with
arXiv:1912.0712
Discontinuous Galerkin Methods for Fisher-Kolmogorov Equation with Application to -Synuclein Spreading in Parkinson's Disease
The spreading of prion proteins is at the basis of brain neurodegeneration.
The paper deals with the numerical modelling of the misfolding process of
-synuclein in Parkinson's disease. We introduce and analyze a
discontinuous Galerkin method for the semi-discrete approximation of the
Fisher-Kolmogorov (FK) equation that can be employed to model the process. We
employ a discontinuous Galerkin method on polygonal and polyhedral grids
(PolyDG) for space discretization, which allows us to accurately simulate the
wavefronts typically observed in the prionic spreading. We prove stability and
a priori error estimates for the semi-discrete formulation. Next, we use a
Crank-Nicolson scheme to advance in time. For the numerical verification of our
numerical model, we first consider a manufactured solution, and then we
consider a case with wavefront propagation in two-dimensional polygonal grids.
Next, we carry out a simulation of -synuclein spreading in a
two-dimensional brain slice in the sagittal plane with a polygonal agglomerated
grid that takes full advantage of the flexibility of PolyDG approximation.
Finally, we present a simulation in a three-dimensional patient-specific brain
geometry reconstructed from magnetic resonance images.Comment: arXiv admin note: text overlap with arXiv:2210.0227
Multigrid algorithms for -discontinuous Galerkin discretizations of elliptic problems
Abstract. We present W-cycle multigrid algorithms for the solution of the linear system of equations arising from a wide class of hp-version discontinuous Galerkin discretizations of elliptic problems. Starting from a classical framework in multigrid analysis, we define a smoothing and an approximation property, which are used to prove the uniform convergence of the W-cycle scheme with respect to the granularity of the grid and the number of levels. The dependence of the convergence rate on the polynomial approximation degree p is also tracked, showing that the contraction factor of the scheme deteriorates with increasing p. A discussion on the effects of employing inherited or non-inherited sublevel solvers is also presented. Numerical experiments confirm the theoretical results. Key words. hp-version discontinuous Galerkin, multigrid algorithms, elliptic problem