Curl-curl Operator Research Articles

An efficient multigrid solver based on a four-color cell-block Gauss-Seidel smoother for 3D magnetotelluric forward modeling

Practical application of 3D magnetotelluric inversion requires efficient forward modeling of electromagnetic (EM) fields in the earth. To resolve realistic 3D structures, large computational domains and extremely large linear systems of equations are required. The iterative solvers, which are almost exclusively used to solve these systems, can be inefficient due to the abundant null space of the curl-curl operator. Multigrid (MG) solvers are considered a potentially efficient technique for solving such problems. However, due to the abundant null solution space and existence of the air layer, MG solvers can still converge slowly or even diverge. We have developed an efficient MG solver for finite-difference frequency-domain EM solution. In this algorithm, the excellent smoothing property of an efficient four-color cell-block Gauss-Seidel (GS) is exploited to remove the short-range errors effectively, and the interpolation and prolongation operators are used to handle the long-range errors. They work as a whole to speed the convergence of our algorithm remarkably. Because all of the nodes for the four-color cell-block GS are grouped into four colors and the edge components attached to different nodes in each color are completely decoupled, this can be used to develop a highly vectorized or parallelized algorithm. Another important property is that our algorithm is locally current divergence free, effectively eliminating spurious solutions in the null space of the curl-curl operator. The accuracy and efficiency of the algorithm are verified by comparing the numerical solutions obtained with our MG solver to those from the biconjugate gradient stabilized solver with different preconditioners based on synthetic models and a model from 3D inversion. Comparisons, in terms of iteration number and computational time, indicate that our algorithm is extremely stable and efficient relative to the other solvers. Our MG algorithm will be suitable for massively parallel computing as well.

Fast Method for Accelerating Convergence of Iterative Partial Differential Equation Solvers by Changing System Matrix to Laplacian Counterpart

In this work, we find that the matrix representing the curl-curl operator in a partial differential equation solver of Maxwell’s equations can be analytically decomposed into a gradient divergence operator and a Laplacian, both of which can be constructed from the mesh information without any need for computation. The curl-curl operator can hence be replaced by the Laplacian to find the divergence-free component of the field solution. The Laplacian is positive definite and well-conditioned. As a result, the convergence of an iterative solution of Maxwell’s equations can be guaranteed, and also significantly accelerated. Based on the finding, we represent the divergence-free component of the unknown field solution by deducting its curl-free component. The curl-free component resides in the nullspace of the curl-curl operator, which is also analytically known from the mesh information no matter it is a regular grid or an unstructured mesh. After the divergence-free component is rapidly solved from a Laplacian counterpart of the original system matrix, the curl-free component can also be solved from a Laplacian matrix, and hence having fast and guaranteed convergence. The total computational cost of the proposed method is simply a small number of sparse matrix-vector multiplications. The proposed method has been successfully applied to solve ill-conditioned onchip, packaging, and antenna radiation problems at both low and high frequencies, involving both inhomogeneous dielectrics and lossy conductors. Numerical experiments have demonstrated its fast and guaranteed convergence, as well as trivial computational cost independent of problem size.

Spectral stability of the curlcurl operator via uniform Gaffney inequalities on perturbed electromagnetic cavities

<abstract><p>We prove spectral stability results for the $ curl curl $ operator subject to electric boundary conditions on a cavity upon boundary perturbations. The cavities are assumed to be sufficiently smooth but we impose weak restrictions on the strength of the perturbations. The methods are of variational type and are based on two main ingredients: the construction of suitable Piola-type transformations between domains and the proof of uniform Gaffney inequalities obtained by means of uniform a priori $ H^2 $-estimates for the Poisson problem of the Dirichlet Laplacian. The uniform a priori estimates are proved by using the results of V. Maz'ya and T. Shaposhnikova based on Sobolev multipliers. Connections to boundary homogenization problems are also indicated.</p></abstract>

A Scalable Approximate Inverse Block Preconditioner for an Incompressible Magnetohydrodynamics Model Problem

We introduce a new approximate inverse preconditioner for a mixed finite element discretization of an incompressible magnetohydrodynamics model problem. The derivation exploits the nullity of the discrete curl-curl operator in the Maxwell subproblem. We show that the inverse of the coefficient matrix contains zero blocks and use discretization considerations to obtain a practical preconditioner based on further sparsification. We demonstrate the viability of our approach with a set of numerical experiments.

Uncountably Many Solutions for Nonlinear Helmholtz and Curl-Curl Equations

Abstract We obtain uncountably many solutions of nonlinear Helmholtz and curl-curl equations on the entire space using a fixed point approach. The constructed solutions are mildly localized as they lie in the essential spectrum of the corresponding linear operator. As a new auxiliary tool a limiting absorption principle for the curl-curl operator is proved.

Spectral analysis and spectral symbol for the 2D curl-curl (stabilized) operator with applications to the related iterative solutions

Spectral analysis and spectral symbol for the 2D curl–curl (stabilized) operator with applications to the related iterative solutions

An a priori error estimate for interior penalty discretizations of the Curl-Curl operator on non-conforming meshes

We prove an a-priori error estimate for regularized Curl-Curl Problems which are discretized by the Interior Penalty/Nitsche’s Method on meshes non-conforming across interfaces. It is shown that the total error can be bounded by the best approximation error which in turn depends on the concrete choice of the approximation space $V_{h}$ . In this work we show that if $V_{h}$ is the space of edge functions of the first kind of order k we can expect (suboptimal) convergence $O(h^{k-1})$ as the mesh is refined. The numerical experiments in (Casagrande et al., SAM Report 2014-32, ETH Zurich, 2014) indicate that this bound is sharp for $k=1$ . Moreover it is shown that the regularization term can be made arbitrarily small without affecting the error in the $\lvert \cdot \rvert_{\mathbf {curl}}$ semi-norm. A numerical experiment shows that the regularization parameter can be chosen in a wide range of values such that, at the same time, the discrete problem remains solvable and the error due to regularization is negligible compared to the discretization error.

A Breather Construction for a Semilinear Curl-Curl Wave Equation with Radially Symmetric Coefficients

We consider the semilinear curl-curl wave equation $$s\left( x \right)\partial _t^2U + \nabla \times \nabla \times + q\left( x \right)U \pm V\left( x \right){\left| U \right|^{p - {\text{ 1}}}}U = 0for\left( {x,t} \right) \in {\mathbb{R}^3} \times \mathbb{R}.$$ For any p < 1 we prove the existence of time-periodic spatially localized real-valued solutions (breathers) both for the + and the - case under slightly different hypotheses. Our solutions are classical solutions that are radially symmetric in space and decay exponentially to 0 as |x| → ∞. Our method is based on the fact that gradient fields of radially symmetric functions are annihilated by the curl-curl operator. Consequently, the semilinear wave equation is reduced to an ODE with r = |x| as a parameter. This ODE can be efficiently analyzed in phase space. As a side effect of our analysis, we obtain not only one but a full continuum of phase-shifted breathers U(x; t + a(x)), where U is a particular breather and a: ℝ3 → ℝ an arbitrary radially symmetric C2-function.

Alternative Method for Making Explicit FDTD Unconditionally Stable

An alternative method is developed to make an explicit FDTD unconditionally stable. In this method, given any time step, we find the modes that cannot be stably simulated by the given time step, and deduct these modes directly from the system matrix (discretized curl-curl operator) before the explicit time marching. By doing so, the original FDTD numerical system is adapted based on the desired time step to rule out the root cause of instability. The resultant explicit FDTD marching is absolutely stable for the given time step no matter how large it is, and irrespective of space step. The accuracy is also guaranteed for time step chosen based on accuracy. Numerical experiments have validated the accuracy, efficiency, and unconditional stability of the proposed new method for making an explicit FDTD unconditionally stable.

A staggered discontinuous Galerkin method for the curl-curl operator

This paper is concerned with a staggered discontinuous Galerkin method for the curl–curl operator arising from the time-harmonic Maxwell equations. One distinctive feature of the method is that the discrete operators preserve the properties of the differential operators. Moreover, the numerical solution automatically satisfies a discrete divergence-free condition. Stability and optimal convergence of the method are analysed. Numerical experiments for smooth and singular solutions are shown to verify the optimal order of convergence. Furthermore, the method is applied to the corresponding eigenvalue problem. Numerical results for rectangular and L-shaped domains show that our method is able to produce nonspurious eigenvalues.

Local Fourier Analysis of Multigrid for the Curl-Curl Equation

We present a local Fourier analysis of multigrid methods for the two-dimensional curl-curl formulation of Maxwell's equations. Both the hybrid smoother proposed by Hiptmair and the overlapping block smoother proposed by Arnold, Falk, and Winther are considered. The key to our approach is the identification of two-dimensional eigenspaces of the discrete curl-curl problem by decoupling the Fourier modes for edges with different orientations. This procedure is used to quantify the smoothing properties of the considered smoothers and the convergence behavior of the multigrid methods. Additionally, we identify the Helmholtz splitting in Fourier space. This allows several well known properties to be recovered in Fourier space, such as the commutation properties of the classical Nedelec prolongator and the equivalence of the curl-curl operator and the vector Laplacian for divergence-free vectors. We show how the approach used in this paper can be generalized to two- and three-dimensional problems in $H$(curl) and $H$(div) and to other types of regular meshes.

Mixed discontinuous Galerkin approximation of the Maxwell operator: The indefinite case

We present and analyze an interior penalty method for the numerical discretization of the indefinite time-harmonic Maxwell equations in mixed form. The method is based on the mixed discretization of the curl-curl operator developed in [Houston et al. , J. Sci. Comp. 22 (2005) 325–356] and can be understood as a non-stabilized variant of the approach proposed in [Perugia et al. , Comput. Methods Appl. Mech. Engrg. 191 (2002) 4675–4697]. We show the well-posedness of this approach and derive optimal a priori error estimates in the energy-norm as well as the L 2 -norm. The theoretical results are confirmed in a series of numerical experiments.

"Stiff" problems in eddy-current theory and the regularization of Maxwell's equations

Large ratios of extreme nonzero conductivities in eddy current problems cause poor conditioning, due to the kernel of the curl-curl operator. Solutions to this problem (wrongly blamed on edge elements by some) are discussed.