Rm Curl Research Articles

Low-Order Preconditioning for the High-Order Finite Element de Rham Complex

In this paper we present a unified framework for constructing spectrally equivalent low-order-refined discretizations for the high-order finite element de Rham complex. This theory covers diffusion problems in $H^1$, $H({\rm curl})$, and $H({\rm div})$, and is based on combining a low-order discretization posed on a refined mesh with a high-order basis for N\'ed\'elec and Raviart-Thomas elements that makes use of the concept of polynomial histopolation (polynomial fitting using prescribed mean values over certain regions). This spectral equivalence, coupled with algebraic multigrid methods constructed using the low-order discretization, results in highly scalable matrix-free preconditioners for high-order finite element problems in the full de Rham complex. Additionally, a new lowest-order (piecewise constant) preconditioner is developed for high-order interior penalty discontinuous Galerkin (DG) discretizations, for which spectral equivalence results and convergence proofs for algebraic multigrid methods are provided. In all cases, the spectral equivalence results are independent of polynomial degree and mesh size; for DG methods, they are also independent of the penalty parameter. These new solvers are flexible and easy to use; any "black-box" preconditioner for low-order problems can be used to create an effective and efficient preconditioner for the corresponding high-order problem. A number of numerical experiments are presented, based on an implmentation in the finite element library MFEM. The theoretical properties of these preconditioners are corroborated, and the flexibility and scalability of the method are demonstrated on a range of challenging three-dimensional problems.

Analytic and geometric properties of dislocation singularities

AbstractThis paper deals with the analysis of the singularities arising from the solutions of the problem${-}\,{\rm Curl\ } F=\mu $, whereFis a 3 × 3 matrix-valuedLp-function ($1\les p<2$) andμa 3 × 3 matrix-valued Radon measure concentrated in a closed loop in Ω ⊂ ℝ3, or in a network of such loops (as, for instance, dislocation clusters as observed in single crystals). In particular, we study the topological nature of such dislocation singularities. It is shown that$F=\nabla u$, the absolutely continuous part of the distributional gradientDuof a vector-valued functionuof special bounded variation. Furthermore,ucan also be seen as a multi-valued field, that is, can be redefined with values in the three-dimensional flat torus 𝕋3and hence is Sobolev-regular away from the singular loops. We then analyse the graphs of such maps represented as currents in Ω × 𝕋3and show that their boundaries can be written in term of the measureμ. Readapting some well-known results for Cartesian currents, we recover closure and compactness properties of the class of maps with bounded curl concentrated on dislocation networks. In the spirit of previous work, we finally give some examples of variational problems where such results provide existence of solutions.

Torsional chiral magnetic effect due to skyrmion textures in a Weyl superfluid He3−A

We investigate torsional chiral magnetic effect (TCME) induced by skyrmion-vortex textures in the A phase of the superfluid $^3$He. In $^3$He-A, Bogoliubov quasiparticles around point nodes behave as Weyl fermions, and the nodal direction represented by the $\ell$-vector may form a spatially modulated texture. $\ell$-textures generate a chiral gauge field and a torsion field directly acting on the chirality of Weyl-Bogoliubov quasiparticles. It has been clarified by G. E. Volovik [Pi'sma Zh. Eksp. Teor. Fiz. {\bf 43}, 428 (1986)] that, if the $\ell$-vector is twisted, the chiral gauge field is responsible for the chiral anomaly, leading to an anomalous current along ${\ell}$. Here we show that, even for non-twisted $\ell$-vector fields, a torsion arising from $\ell$-textures brings about contributions to the equilibrium currents of Weyl-Bogoliubov quasiparticles along ${\rm curl}{\ell}$. This implies that while the anomalous current appears only for the twisted (Bloch-type) skyrmion of the $\ell$-vector, the extra mass current due to TCME always exists regardless of the skyrmion type. Solving the Bogoliubov-de Gennes equation, we demonstrate that both Bloch-type and N\'{e}el-type skyrmions induce chiral fermion states with spectral asymmetry, and possess spatially inhomogeneous structures of Weyl bands in the real coordinate space. Furthermore, we discuss the contributions of Weyl-Bogoliubov quasiparticles and continuum states to the mass current density in the vicinity of the topological phase transition. In the weak coupling limit, continuum states give rise to backflow to the mass current generated by Weyl-Bogoliubov quasiparticles, which makes a non-negligible contribution to the orbital angular momentum. As the topological transition is approached, the mass current density is governed by the contribution of continuum states.

Stream function formulation of surface Stokes equations

AbstractIn this paper we present a derivation of the surface Helmholtz decomposition, discuss its relation to the surface Hodge decomposition and derive a well-posed stream function formulation of a class of surface Stokes problems. We consider a $C^2$ connected (not necessarily simply connected) oriented hypersurface $\varGamma \subset \mathbb{R}^3$ without boundary. The surface gradient, divergence, curl and Laplace operators are defined in terms of the standard differential operators of the ambient Euclidean space $\mathbb{R}^3$. These representations are very convenient for the implementation of numerical methods for surface partial differential equations. We introduce surface $\mathbf H({\mathop{\rm div}}_{\varGamma})$ and $\mathbf H({\mathop{\rm curl}}_{\varGamma})$ spaces and derive useful properties of these spaces. A main result of the paper is the derivation of the Helmholtz decomposition, in terms of these surface differential operators, based on elementary differential calculus. As a corollary of this decomposition we obtain that for a simply connected surface to every tangential divergence-free velocity field there corresponds a unique scalar stream function. Using this result the variational form of the surface Stokes equation can be reformulated as a well-posed variational formulation of a fourth-order equation for the stream function. The latter can be rewritten as two coupled second-order equations, which form the basis for a finite element discretization. A particular finite element method is explained and the results of a numerical experiment with this method are presented.

High-dimensional finite elements for multiscale Maxwell-type equations

We consider multiscale Maxwell-type equations in a domain |$D\subset\mathbb{R}^d$| (⁠|$d=2,3$|⁠), which depend on |$n$| microscopic scales. Using multiscale convergence, we derive the multiscale homogenized problem, which is posed in |$\mathbb{R}^{(n+1)d}$|⁠. Solving it, we get all the necessary macroscopic and microscopic information. Sparse tensor product finite elements (FEs) are employed, using edge FEs. The method achieves a required level of accuracy with essentially an optimal number of degrees of freedom, which, apart from a multiplying logarithmic term, is equal to that for solving a problem in |$\mathbb{R}^d$|⁠. Numerical correctors are constructed from the FE solutions. In the two-scale case, an explicit homogenization error is deduced. To get this error, the standard procedure in the homogenization literature requires the solution |$u_0$| of the homogenized problem to belong to |$H^1({\rm curl\,},D)$|⁠. However, in polygonal domains, |$u_0$| belongs only to a weaker regularity space |$H^s({\rm curl\,},D)$| for |$0<s<1$|⁠. We derive a homogenization error estimate for this case. Though we prove the result for two-scale Maxwell-type equations, the approach works verbatim for elliptic and elasticity problems when the solution to the homogenized equation belongs to |$H^{1+s}(D)$| (standard procedure requires |$H^2(D)$| regularity). This homogenization error estimate is new in the literature. Thus, for two-scale problems, an explicit error for the numerical corrector is obtained; it is of the order of the sum of the homogenization error and the FE error. For the case of more than two scales, we construct a numerical corrector, albeit without a rate of convergence, as such a homogenization error is not available. Numerical experiments confirm the theoretical results.

Ground states of a nonlinear curl-curl problem in cylindrically symmetric media

We consider the nonlinear curl-curl problem $${\nabla\times\nabla\times U + V(x) U= \Gamma(x)|U|^{p-1}U}$$ in $${\mathbb{R}^3}$$ related to the Kerr nonlinear Maxwell equations for fully localized monochromatic fields. We search for solutions as minimizers (ground states) of the corresponding energy functional defined on subspaces (defocusing case) or natural constraints (focusing case) of $${H({\rm curl};\mathbb{R}^3)}$$ . Under a cylindrical symmetry assumption corresponding to a photonic fiber geometry on the functions V and $${\Gamma}$$ the variational problem can be posed in a symmetric subspace of $${H({\rm curl};\mathbb{R}^3)}$$ . For a defocusing case $${{\rm sup} \Gamma < 0}$$ with large negative values of $${\Gamma}$$ at infinity we obtain ground states by the direct minimization method. For the focusing case $${{\rm inf} \Gamma > 0}$$ the concentration compactness principle produces ground states under the assumption that zero lies outside the spectrum of the linear operator $${\nabla \times \nabla \times +V(x)}$$ . Examples of cylindrically symmetric functions V are provided for which this holds.

A Mixed Formulation for the Brinkman Problem

The Brinkman model is a unified law governing the flow of a viscous fluid in an inhomogeneous medium, where fractures, bubbles, or channels alternate inside a porous matrix. In this work, we explore a novel mixed formulation of the Brinkman problem based on the Hodge decomposition of the vector Laplacian. Introducing the flow's vorticity as an additional unknown, this formulation allows for a uniformly stable and conforming discretization by standard finite elements (Nedelec, Raviart--Thomas, piecewise discontinuous). A priori error estimates for the discretization error in the $H({\rm curl}; \Omega)-H({\rm div}; \Omega)-L^2(\Omega)$ norm of the solution, which are optimal with respect to the approximation properties of finite element spaces, are obtained. The theoretical results are illustrated with numerical experiments. Finally, the proposed formulation allows for a scalable block diagonal preconditioner which takes advantage of the auxiliary space algebraic multigrid solvers for $H({\rm curl})$ and $H...

A Block-Diagonal Algebraic Multigrid Preconditioner for the Brinkman Problem

The Brinkman model is a unified law governing the flow of a viscous fluid in cavity (Stokes equations) and porous media (Darcy equations). In this work, we explore a novel mixed formulation of the Brinkman problem by introducing the flow's vorticity as an additional unknown. This formulation allows for a uniformly stable and conforming discretization by standard finite element (Nedelec, Raviart--Thomas, discontinuous piecewise polynomials). Based on the stability analysis of the problem in the $H({\rm curl})-H({\rm div})-L^2$ norms [P. S. Vassilevski and U. Villa, A mixed formulation for the Brinkman problem, SIAM J. Numer. Anal., submitted], we study a scalable block-diagonal preconditioner which is provably optimal in the constant coefficient case. Such a preconditioner takes advantage of the parallel auxiliary space AMG solvers for $H({\rm curl})$ and $H({\rm div})$ problems available in HYPRE [hypre: High Performance Preconditioners, http://www.llnl.gov/CASC/hypre/]. The theoretical results are illust...

A Delta-Regularization Finite Element Method for a Double Curl Problem with Divergence-Free Constraint

To deal with the divergence-free constraint in a double curl problem, ${\rm curl\,} \mu^{-1} {\rm curl\,} u=f$ and ${\rm div\,} \varepsilon u=0$ in $\Omega$, where $\mu$ and $\varepsilon$ represent the physical properties of the materials occupying $\Omega$, we develop a $\delta$-regularization method, ${\rm curl\,} \mu^{-1} {\rm curl\,} u_\delta +\delta \varepsilon u_\delta=f$, to completely ignore the divergence-free constraint ${\rm div\,} \varepsilon u=0$. We show that $u_\delta$ converges to $u$ in $H({\rm curl\,};\Omega)$ norm as $\delta\rightarrow 0$. The edge finite element method is then analyzed for solving $u_\delta$. With the finite element solution $u_{\delta,h}$, a quasioptimal error bound in the $H({\rm curl\,};\Omega)$ norm is obtained between $u$ and $u_{\delta,h}$, including a uniform (with respect to $\delta$) stability of $u_{\delta,h}$ in the $H({\rm curl\,};\Omega)$ norm. All the theoretical analysis is done in a general setting, where $\mu$ and $\varepsilon$ may be discontinuous, anisotropic, and inhomogeneous, and the solution may have a very low piecewise regularity on each material subdomain $\Omega_j$ with $u,{\rm curl\,} u\in (H^r(\Omega_j))^3$ for some $0<r<1$, where $r$ may not be greater than $1/2$. To establish the uniform stability and the error bound for $r\le 1/2$, we have respectively developed a new theory for the $\mathscr{K}_h$ ellipticity (related to mixed methods) and a new theory for the Fortin interpolation operator. Numerical results confirm the theory.

Optimal Error Estimation for ${\bfH}({\rmcurl})$-Conformingp-Interpolation in Two Dimensions

In this paper we prove an optimal error estimate for the ${\bf H}({\rm curl})$-conforming projection-based p-interpolation operator introduced in [L. Demkowicz and I. Babuška, SIAM J. Numer. Anal., 41 (2003), pp. 1195–1208]. This result is proved on the reference element (either triangle or square) K for regular vector fields in ${\bf H}^r({\rm curl},K)$ with arbitrary $r >0$. The formulation of the result in the ${\bf H}({\rm div})$-conforming setting, which is relevant for the analysis of high-order boundary element approximations for Maxwell's equations, is provided as well.

The Discrete First-Order System Least Squares: The Second-Order Elliptic Boundary Value Problem

In [Z. Cai, T. Manteuffel, and S. F. McCormick, SIAM J. Numer. Anal., 34 (1997), pp. 425--454], an L2 -norm version of first-order system least squares (FOSLS) was developed for scalar second-order elliptic partial differential equations. A limitation of this approach is the requirement of sufficient smoothness of the original problem, which is used for the equivalence of spaces between (H1 )d and $H(\div)\cap H({\rm curl})$-type, where d=2 or 3 is the dimension. By directly approximating $H(\div)\cap H({\rm curl})$-type space based on the Helmholtz decomposition, this paper develops a discrete FOSLS approach in two dimensions. Under general assumptions, we establish error estimates in the L2 and H1 norms for the vector and scalar variables, respectively. Such error estimates are optimal with respect to the required regularity of the solution. A preconditioner for the algebraic system arising from this approach is also considered.