49 research outputs found
On stability of discretizations of the Helmholtz equation (extended version)
We review the stability properties of several discretizations of the
Helmholtz equation at large wavenumbers. For a model problem in a polygon, a
complete -explicit stability (including -explicit stability of the
continuous problem) and convergence theory for high order finite element
methods is developed. In particular, quasi-optimality is shown for a fixed
number of degrees of freedom per wavelength if the mesh size and the
approximation order are selected such that is sufficiently small and
, and, additionally, appropriate mesh refinement is used near
the vertices. We also review the stability properties of two classes of
numerical schemes that use piecewise solutions of the homogeneous Helmholtz
equation, namely, Least Squares methods and Discontinuous Galerkin (DG)
methods. The latter includes the Ultra Weak Variational Formulation
The Boundary Convergence of the Steady Zero-Temperature-Driven Hard Spheres
We study the fundamental problem of two gas species in two dimensional
velocity space whose molecules collide as hard circles in the presence of a
flat boundary and with dependence on only one space dimension. The case of
three-dimensional velocity space is a generalization. More speciffically the
linear problem arising when the second gas dominates as a flow with constant
velocity (and hence zero temperature) is considered. The boundary condition
adopted consists of prescribing the outgoing velocity distribution at the wall.
It is discovered that the presence of the boundary under general assumptions on
the outgoing distribution ensures the convergence of a series of path integrals
and thus a convenient representation for the solution is obtained.Comment: 13 pages, 2 figure
A space–time Trefftz discontinuous Galerkin method for the acoustic wave equation in first-order formulation
We introduce a space–time Trefftz discontinuous Galerkin method for the first-order transient acoustic wave equations in arbitrary space dimensions, extending the one-dimensional scheme of Kretzschmar et al. (IMA J Numer Anal 36:1599–1635, 2016). Test and trial discrete functions are space–time piecewise polynomial solutions of the wave equations. We prove well-posedness and a priori error bounds in both skeleton-based and mesh-independent norms. The space–time formulation corresponds to an implicit time-stepping scheme, if posed on meshes partitioned in time slabs, or to an explicit scheme, if posed on “tent-pitched” meshes. We describe two Trefftz polynomial discrete spaces, introduce bases for them and prove optimal, high-order h-convergence bounds