5,191 research outputs found
Numerical analysis of a transmission problem with Signorini contact using mixed-FEM and BEM
© EDP Sciences, SMAI 2011This paper is concerned with the dual formulation of the interface problem consisting of a linear partial differential equation with variable coefficients in some bounded Lipschitz domain Ω in
Rn (n ≥ 2) and the Laplace equation with some radiation condition in the unbounded exterior domain Ωc := Rn\ ̄Ω. The two problems are coupled by transmission and Signorini contact conditions on the interface Γ = ∂Ω. The exterior part of the interface problem is rewritten using a Neumann to Dirichlet mapping (NtD) given in terms of boundary integral operators. The resulting variational formulation becomes a variational inequality with a linear operator. Then we treat the corresponding numerical scheme and discuss an approximation of the NtD mapping with an appropriate discretization of the inverse Poincar´e-Steklov operator. In particular, assuming some abstract approximation properties and a discrete inf-sup condition, we show unique solvability of the discrete scheme and obtain the corresponding a-priori error estimate. Next, we prove that these assumptions are satisfied with Raviart- Thomas elements and piecewise constants in Ω, and continuous piecewise linear functions on Γ. We suggest a solver based on a modified Uzawa algorithm and show convergence. Finally we present some numerical results illustrating our theory
Weak Form of Stokes-Dirac Structures and Geometric Discretization of Port-Hamiltonian Systems
We present the mixed Galerkin discretization of distributed parameter
port-Hamiltonian systems. On the prototypical example of hyperbolic systems of
two conservation laws in arbitrary spatial dimension, we derive the main
contributions: (i) A weak formulation of the underlying geometric
(Stokes-Dirac) structure with a segmented boundary according to the causality
of the boundary ports. (ii) The geometric approximation of the Stokes-Dirac
structure by a finite-dimensional Dirac structure is realized using a mixed
Galerkin approach and power-preserving linear maps, which define minimal
discrete power variables. (iii) With a consistent approximation of the
Hamiltonian, we obtain finite-dimensional port-Hamiltonian state space models.
By the degrees of freedom in the power-preserving maps, the resulting family of
structure-preserving schemes allows for trade-offs between centered
approximations and upwinding. We illustrate the method on the example of
Whitney finite elements on a 2D simplicial triangulation and compare the
eigenvalue approximation in 1D with a related approach.Comment: Copyright 2018. This manuscript version is made available under the
CC-BY-NC-ND 4.0 license http://creativecommons.org/licenses/by-nc-nd/4.0
Dual approach to circuit quantization using loop charges
The conventional approach to circuit quantization is based on node fluxes and
traces the motion of node charges on the islands of the circuit. However, for
some devices, the relevant physics can be best described by the motion of
polarization charges over the branches of the circuit that are in general
related to the node charges in a highly nonlocal way. Here, we present a
method, dual to the conventional approach, for quantizing planar circuits in
terms of loop charges. In this way, the polarization charges are directly
obtained as the differences of the two loop charges on the neighboring loops.
The loop charges trace the motion of fluxes through the circuit loops. We show
that loop charges yield a simple description of the flux transport across
phase-slip junctions. We outline a concrete construction of circuits based on
phase-slip junctions that are electromagnetically dual to arbitrary planar
Josephson junction circuits. We argue that loop charges also yield a simple
description of the flux transport in conventional Josephson junctions shunted
by large impedances. We show that a mixed circuit description in terms of node
fluxes and loop charges yields an insight into the flux decompactification of a
Josephson junction shunted by an inductor. As an application, we show that the
fluxonium qubit is well approximated as a phase-slip junction for the
experimentally relevant parameters. Moreover, we argue that the - qubit
is effectively the dual of a Majorana Josephson junction.Comment: 20 pages, 11 figures. Version accepted for publication in PRB.
Changes: introduction has become less technical and an example for the
inclusion of offset charges has been adde
A low-order mixed finite element method for a class of quasi-Newtonian Stokes flows. Part I: a priori error analysis
[Abstract] We present a mixed finite element method for a class of non-linear Stokes models arising in quasi-Newtonian fluids.
Our results include, as a by-product, a new mixed scheme for the linear Stokes equation. The approach is based on the
introduction of both the flux and the tensor gradient of the velocity as further unknowns, which yields a twofold saddle
point operator equation as the resulting variational formulation. We prove that the continuous and discrete formulations
are well posed, and derive the associated a priori error analysis. The corresponding Galerkin scheme is defined
by using piecewise constant functions and Raviart–Thomas spaces of lowest order
- …