Curl System Research Articles

An [formula omitted]- primal–dual weak Galerkin method for div–curl systems

This paper presents a new Lp-primal–dual weak Galerkin (PDWG) finite element method for the div–curl system with the normal boundary condition for p>1. Two crucial features for the proposed Lp-PDWG finite element scheme are as follows: (1) it offers an accurate and reliable numerical solution to the div–curl system under the low Wα,p-regularity (α>0) assumption for the exact solution; (2) it offers an effective approximation of the normal harmonic vector fields on domains with complex topology. An optimal order error estimate is established in the Lq-norm for the primal variable where 1p+1q=1. A series of numerical experiments are presented to demonstrate the performance of the proposed Lp-PDWG algorithm.

Div–Curl System with Potential and Maxwell–Stokes System with Natural Boundary Condition

This paper concerns the boundary value problems of div–curl system with potential, and those of the Maxwell–Stokes system under natural boundary condition. One of the main features of these systems is that the equations contain an unknown potential term, which represents the effect of domain topology. Solvability of these systems depends not only on the structure of the equations and the type of the boundary conditions, but also on the domain topology. We show existence of solutions of these systems by using variational methods together with a modified de Rham lemma, and by methods of compact operators. Regularity of the solutions and the eigenvalue problem are also discussed.

On a Quasilinear Parabolic Curl System Motivated by Time Evolution of Meissner States of Superconductors

On a Quasilinear Parabolic Curl System Motivated by Time Evolution of Meissner States of Superconductors

Maxwell–Stokes system with $$L^2$$ boundary data and Div–Curl system with potential

This paper concerns the boundary value problems of two partial differential systems involving the operator curl and containing an unknown potential, and under boundary conditions with $$L^2$$ boundary data. The first one is the Maxwell–Stokes system. We study solvability of both linear and semilinear Maxwell–Stokes systems under either the Dirichlet boundary condition or the natural boundary condition, and examine regularity of the solutions. The second one is the div–curl system with potential, and we derive solvability and regularity under the Dirichlet boundary condition.

A deformation-curl-Poisson decomposition to the three dimensional steady Euler-Poisson system with applications

In this paper, we provide a deformation-curl-Poisson decomposition for the three dimensional steady Euler-Poisson system. By using the Bernoulli's law, we rewrite the density equation as a Frobenius inner product of a symmetric matrix and the deformation matrix with an additional term reflecting the electrostatic effect. The vorticity will be solved by a transport equation with another two algebraic equations, hence one obtains a nonlinear system consisting of the deformation equation, the curl system and the Poisson equation, which is elliptic in the sense of Agmon, Dougalis and Nirenberg. As an application, we establish the structural stability of 1-D subsonic background solutions with multidimensional perturbations on the entrance and exit of a rectangular cylinder. The key ingredient of the analysis lies in some a priori estimates for the deformation-curl-Poisson system with tangential or normal boundary conditions for the velocity.

A deformation-curl decomposition for three dimensional steady Euler equations

In this paper, we provide a deformation-curl decomposition of three dimensional steady Euler equation. The key issue in this new decomposition is based on a simple observation that the density can be represented as a function of the Bernoullis function, the entropy and the speed, and hence the density equation can be rewritten as a Frobenius inner product of a symmetric matrix with the deformation matrix: $A(\boldsymbol{u}):~\mathcal{D}(\boldsymbol{u})=0$, where $\mathcal{D}(\boldsymbol{u})=(\frac12(\partial_j~u_k+~\partial_k~u_j))_{j,k=1}^3$ is the deformation matrix. Furthermore, employing the momentum equations, we find the vorticity can be resolved by a transport equation with another two algebraic equations involving the Bernoullis function and entropy.We also obtain the velocity by solving the deformation and curl system. The most important advantage of this decomposition is the velocity, the Bernoullis function and entropy we obtained share the same regularity as the pressure or density. Based on this decomposition, we prove the existence and uniqueness of subsonic flows in a finitely long rectangular nozzle by prescribing suitable physical boundary conditions.

Meissner states of type II superconductors

This paper concerns mathematical theory of Meissner states of a bulk superconductor of type $I\!I$, which occupies a bounded domain $\Omega$ in $\Bbb R^3$ and is subjected to an applied magnetic field below the critical field $H_{S}$. A Meissner state is described by a solution $(f,\mathbf A)$ of a nonlinear partial differential system called Meissner system, where $f$ is a positive function on $\Omega$ which is equal to the modulus of the order parameter, and $\mathbf A$ is the magnetic potential defined on the entire space such that the inner trace of the normal component on the domain boundary $\partial\Omega$ vanishes. Such a solution is called a Meissner solution. Various properties of the Meissner solutions are examined, including regularity, classification and asymptotic behavior for large value of the Ginzburg-Landau parameter $\kappa$. It is shown that the Meissner solution is smooth in $\Omega$, however the regularity of the magnetic potential outside $\Omega$ can be rather poor. This observation leads to the ides of decomposition of the Meissner system into two problems, a boundary value problem in $\Omega$ and an exterior problem outside of $\Omega$. We show that the solutions of the boundary value problem with fixed boundary data converges uniformly on $\Omega$ as $\kappa$ tends to $\infty$, where the limit field of the magnetic potential is a solution of a nonlinear curl system. This indicates that, the magnetic potential part $\mathbf A$ of the solution $(f,\mathbf A)$ of the Meissner system, which has same tangential component of $curl \mathbf A$ on $\partial\Omega$, converges to a solution of the curl system as $\kappa$ increases to infinity, which verifies that the curl system is indeed the correct limit of the Meissner system in the case of three dimensions.

Variational problem involving operator curl associated with p -curl system

Variational problem involving operator curl associated with p -curl system

Instantaneous PIV/PTV-based pressure gradient estimation: a framework for error analysis and correction

A framework for the exact determination of the pressure gradient estimation error in incompressible flows given erroneous velocimetry data is derived which relies on the calculation of the curl and divergence of the pressure gradient error over the domain and then the solution of a div–curl system to reconstruct the pressure gradient error field. In practice, boundary conditions for the div–curl system are unknown, and the divergence of the pressure gradient error requires approximation. The effect of zero pressure gradient error boundary conditions and approximating the divergence are evaluated using three flow cases: (1) a stationary Taylor vortex; (2) an advecting Lamb–Oseen vortex near a boundary; and (3) direct numerical simulation of the turbulent wake of a circular cylinder. The results show that the exact form of the pressure gradient error field reconstruction converges onto the exact values, within truncation and round-off errors, except for a small flow field region near the domain boundaries. It is also shown that the approximation for the divergence of the pressure gradient error field retains the fidelity of the reconstruction, even when velocity field errors are generated with substantial spatial variation. In addition to the utility of the proposed technique to improve the accuracy of pressure estimates, the reconstructed error fields provide spatially resolved estimates for instantaneous PIV/PTV-based pressure error.

Three dimensional steady subsonic Euler flows in bounded nozzles

The existence and uniqueness of three dimensional steady subsonic Euler flows in rectangular nozzles were obtained when prescribing normal component of momentum at both the entrance and exit. If, in addition, the normal component of the voriticity and the variation of Bernoulli's function at the entrance are both zero, then there exists a unique subsonic potential flow when the magnitude of the normal component of the momentum is less than a critical number. As the magnitude of the normal component of the momentum approaches the critical number, the associated flows converge to a subsonic–sonic flow. Furthermore, when the normal component of vorticity and the variation of Bernoulli function are both small, the existence and uniqueness of subsonic Euler flows with non-zero vorticity are established. The proof of these results is based on a new formulation for the Euler system, a priori estimate for nonlinear elliptic equations with nonlinear boundary conditions, detailed study for a linear div–curl system, and delicate estimate for the transport equations.

Multilevel discretization of symmetric saddle point systems without the discrete LBB condition

Using an inexact Uzawa algorithm at the continuous level, we study the convergence of multilevel algorithms for solving saddle-point problems. The discrete stability Ladyshenskaya–Babušca–Brezzi (LBB) condition does not have to be satisfied. The algorithms are based on the existence of a multilevel sequence of nested approximation spaces for the constrained variable. The main idea is to maintain an accurate representation of the residual associated with the main equation at each step of the inexact Uzawa algorithm at the continuous level. The residual representation is approximated by a Galerkin projection. Whenever a sufficient condition for the accuracy of the representation fails to be satisfied, the representation of the residual is projected on the next (larger) space available in the prescribed multilevel sequence. Numerical results supporting the efficiency of the algorithms are presented for the Stokes equations and a div–curl system.

Finite element approximation of field dislocation mechanics

A tool for studying links between continuum plasticity and dislocation theory within a field framework is presented. A finite element implementation of the geometrically linear version of a recently proposed theory of field dislocation mechanics (J. Mech. Phys. Solids 49 (2001) 761; Proc. Roy. Soc. 459 (2003) 1343; J. Mech. Phys. Solids 52 (2004) 301) represents the main idea behind the tool. The constitutive ingredients of the theory under consideration are simply elasticity and a specification of dislocation velocity and nucleation. The set of equations to be approximated are non-standard in the context of solid mechanics applications. It comprises the standard second-order equilibrium equations, a first-order div–curl system for the elastic incompatibility, and a first-order, wave-propagative system for the evolution of dislocation density. The latter two sets of equations require special treatment as the standard Galerkin method is not adequate, and are solved utilizing a least-squares finite element strategy. The implementation is validated against analytical results of the classical elastic theory of dislocations and analytical results of the theory itself. Elastic stress fields of dislocation distributions in generally anisotropic media of finite extent, deviation from elastic response, yield-drop, and back-stress are shown to be natural consequences of the model. The development of inhomogeneity, from homogeneous initial conditions and boundary conditions corresponding to homogeneous deformation in conventional plasticity, is also demonstrated. To our knowledge, this work represents the first computational implementation of a theory of dislocation mechanics where no analytical results, singular solutions in particular, are required to formulate the implementation. In particular, a part of the work is the first finite element implementation of Kröner's linear elastic theory of continuously distributed dislocations in its full generality.

New estimates for the Laplacian, the div–curl, and related Hodge systems

We establish new estimates for the Laplacian, the div–curl system, and more general Hodge systems in arbitrary dimension, with an application to minimizers of the Ginzburg–Landau energy. To cite this article: J. Bourgain, H. Brezis, C. R. Acad. Sci. Paris, Ser. I 338 (2004).

Shear velocity profiles associated with auroral curls

Optical observations using high‐resolution television cameras frequently show that auroral curls are associated with shear velocities in the apparent optical flow. The present study examines in detail one particular curl system event which happened to yield sufficient resolution to determine the fine structure of velocity and vorticity profiles by means of a new analysis technique. Those observations of curl system evolution are contrasted with large velocity shear events where small‐scale quasiperiodic distortions were subject to sudden decay rather than development into vortices. The results are discussed in light of an electrostatic picture of auroral acceleration and the Kelvin‐Helmholtz instability model. We suggest that the latter cannot fully explain the nonlinear phase of the observed curl system event and that curl models should take auroral acceleration processes into account.

The Origin of Spurious Solutions in Computational Electromagnetics

It is commonly believed that the divergence equations in the Maxwell equations are “redundant” for transient and timeharmonic problems, therefore most of the numerical methods in computational electromagnetics solve only two first-order curl equations or the second-order curl–curl equations. This misconception is the true origin of spurious modes and inaccurate solutions in computational electromagnetics. By studying the div–curl system this paper clarifies that the first-order Maxwell equations are not “overdetermined,” and the divergence equations must always be included to maintain the ellipticity of the system in the space domain, to guarantee the uniqueness of the solution and the accuracy of the numerical methods, and to eliminate the infinitely degenerate eigenvalue. This paper shows that the common derivation and usage of the second-order curl–curl equations are incorrect and that the solution of Helmholtz equations needs the divergence condition to be enforced on an associated part of the boundary. The div–curl method and the least-squares method introduced in this paper provide rigorous derivation of the equivalent second-order Maxwell equations and their boundary conditions. The node-based least-squares finite element method (LSFEM) is recommended for solving the first-order full Maxwell equations directly. Examples of the numerical solutions by LSFEM are given to demonstrate that the LSFEM is free of spurious solutions.