32 research outputs found
Uniform -bounds for energy-conserving higher-order time integrators for the Gross-Pitaevskii equation with rotation
In this paper, we consider an energy-conserving continuous Galerkin
discretization of the Gross-Pitaevskii equation with a magnetic trapping
potential and a stirring potential for angular momentum rotation. The
discretization is based on finite elements in space and time and allows for
arbitrary polynomial orders. It was first analyzed in [O. Karakashian, C.
Makridakis; SIAM J. Numer. Anal. 36(6):1779-1807, 1999] in the absence of
potential terms and corresponding a priori error estimates were derived in 2D.
In this work we revisit the approach in the generalized setting of the
Gross-Pitaevskii equation with rotation and we prove uniform -bounds
for the corresponding numerical approximations in 2D and 3D without coupling
conditions between the spatial mesh size and the time step size. With this
result at hand, we are in particular able to extend the previous error
estimates to the 3D setting while avoiding artificial CFL conditions
HDGlab: An Open-Source Implementation of the Hybridisable Discontinuous Galerkin Method in MATLAB
This paper presents HDGlab, an open source MATLAB implementation of the hybridisable discontinuous Galerkin (HDG) method. The main goal is to provide a detailed description of both the HDG method for elliptic problems and its implementation available in HDGlab. Ultimately, this is expected to make this relatively new advanced discretisation method more accessible to the computational engineering community. HDGlab presents some features not available in other implementations of the HDG method that can be found in the free domain. First, it implements high-order polynomial shape functions up to degree nine, with both equally-spaced and Fekete nodal distributions. Second, it supports curved isoparametric simplicial elements in two and three dimensions. Third, it supports non-uniform degree polynomial approximations and it provides a flexible structure to devise degree adaptivity strategies. Finally, an interface with the open-source high-order mesh generator Gmsh is provided to facilitate its application to practical engineering problems
The DPG-star method
This article introduces the DPG-star (from now on, denoted DPG) finite
element method. It is a method that is in some sense dual to the discontinuous
Petrov-Galerkin (DPG) method. The DPG methodology can be viewed as a means to
solve an overdetermined discretization of a boundary value problem. In the same
vein, the DPG methodology is a means to solve an underdetermined
discretization. These two viewpoints are developed by embedding the same
operator equation into two different saddle-point problems. The analyses of the
two problems have many common elements. Comparison to other methods in the
literature round out the newly garnered perspective. Notably, DPG and DPG
methods can be seen as generalizations of and
least-squares methods, respectively. A priori error analysis and a posteriori
error control for the DPG method are considered in detail. Reports of
several numerical experiments are provided which demonstrate the essential
features of the new method. A notable difference between the results from the
DPG and DPG analyses is that the convergence rates of the former are
limited by the regularity of an extraneous Lagrange multiplier variable
A space-time continuous finite element method for 2D viscoelastic wave equation
International audienceA widespread approach to software service analysis uses session types. Very different type theories for binary and multiparty protocols have been developed; establishing precise connections between them remains an open problem. We present the first formal relation between two existing theories of binary and multiparty session types: a binary system rooted in linear logic, and a multiparty system based on automata theory. Our results enable the analysis of multiparty protocols using a (much simpler) type theory for binary protocols, ensuring protocol fidelity and deadlock-freedom. As an application, we offer the first theory of multiparty session types with behavioral genericity. This theory is natural and powerful; its analysis techniques reuse results for binary session types
Quantum Relaxation Method for Linear Systems in Finite Element Problems
Quantum linear system algorithms (QLSAs) for gate-based quantum computing can
provide exponential speedups for linear systems of equations. The growth of the
condition number with problem size for a system of equations arising from a
finite element discretization inhibits the direct application of QLSAs for a
speedup. Furthermore, QLSAs cannot use an approximate solution or initial guess
to output an improved solution. Here, we present Quantum Relaxation for Linear
System (qRLS), as an iterative approach for gate-based quantum computers by
embedding linear stationary iterations into a larger block linear system. The
block linear system is positive-definite and its condition number scales
linearly with the number of iterations independent of the size and condition
number of the original system, effectively managing the condition number of the
finite element problem. The well-conditioned system enables a practical
iterative solution of finite element problems using the state-of-the-art
Quantum Signal Processing (QSP) variant of QLSAs. Using positive-definite QLSAs
l iterations can be performed in O(\sqrt{l}) time, which is unattainable on
classical computers. The complexity of the iterations scales favorably compared
to classical architectures due to solution time scaling independent of system
size with O(\log(N)) qubits, an exponential improvement opening a new paradigm
for iterative finite element solutions on quantum hardware