In this paper, we propose an $H(\text{curl}^2)$-conforming quadrilateral spectral element method to solve quad-curl problems. Starting with generalized Jacobi polynomials, we first introduce quasi-orthogonal polynomial systems for vector fields over rectangles. $H(\text{curl}^2)$-conforming elements over arbitrary convex quadrilaterals are then constructed explicitly in a hierarchical pattern using the contravariant transform together with the bilinear mapping from the reference square onto each quadrilateral. It is astonishing that both the simplest rectangular and quadrilateral spectral elements possess only 8 degrees of freedom on each physical element. In the sequel, we propose our $H(\text{curl}^2)$-conforming quadrilateral spectral element approximation based on the mixed weak formulation to solve the quad-curl equation and its eigenvalue problem. Numerical results show the effectiveness and efficiency of our method.
We construct conforming finite elements for the spaces $H(\text{sym}\,\text{Curl})$ and $H(\text{dev}\,\text{sym}\,\text{Curl})$. Those are spaces of matrix-valued functions with symmetric or deviatoric-symmetric $\text{Curl}$ in a Lebesgue space, and they appear in various models of nonstandard solid mechanics. The finite elements are not $H(\text{Curl})$-conforming. We show the construction, prove conformity and unisolvence, and point out optimal approximation error bounds.
In [24], we proposed H(curl^2)-conforming elements on both a triangle and a rectangle. This family of elements provides a brand new method to solve the quad-curl problem in 2 dimensions. In this paper, we turn our focus to 3 dimensions and construct an H(curl^2)-conforming tetrahedral finite element. The newly proposed element has been proved to have the optimal interpolation error estimate. Having tetrahedral elements, we can solve the quad-curl problem in any Lipschitz domain by conforming finite element method. We also provide several numerical examples of using our element to solve the qua…
We construct smooth finite element de Rham complexes in two space dimensions. This leads to three families of curl-curl conforming finite elements, two of which contain two existing families. The simplest triangular and rectangular finite elements have only 6 and 8 degrees of freedom, respectively. Numerical experiments for each family demonstrate the convergence and efficiency of the elements for solving the quad-curl problem.
$\bf H(\mathrm{curl}^2)$ conforming element for Maxwell's transmission eigenvalue problem using fixed-point approach
Beijing Computational Science Research Center（北京计算科学研究中心）；School of Mathematical Sciences, Guizhou Normal University（贵州师范大学数学科学学院）；Department of Mathematics, Wayne State University（韦恩州立大学数学系）
High-order entropy-stable discontinuous Galerkin (DG) methods for nonlinear conservation laws reproduce a discrete entropy inequality by combining entropy conservative finite volume fluxes with summation-by-parts (SBP) discretization matrices. In the DG context, on tensor product (quadrilateral and hexahedral) elements, SBP matrices are typically constructed by collocating at Lobatto quadrature points. Recent work has extended the construction of entropy-stable DG schemes to collocation at more accurate Gauss quadrature points. In this work, we extend entropy-stable Gauss collocation schemes t…
Dipartimento di Matematica, Università di Milano（米兰大学数学系）；IMATI del CNR（意大利国家研究委员会应用数学研究所）；IUSS（高等研究大学）；Dipartimento di Matematica, Università di Pavia（帕维亚大学数学系）；Dipartimento di Matematica e Applicazioni, Università di Milano-Bicocca（米兰比可卡大学数学与应用科学系）
Novel $H^\mathrm{dev}(\mathrm{Curl})$-conforming elements on regular triangulations and Clough--Tocher splits for the planar relaxed micromorphic model
We present a local construction of H(curl)-conforming piecewise polynomials satisfying a prescribed curl constraint. We start from a piecewise polynomial not contained in the H(curl) space but satisfying a suitable orthogonality property. The procedure employs minimizations in vertex patches and the outcome is, up to a generic constant independent of the underlying polynomial degree, as accurate as the best-approximations over the entire local versions of H(curl). This allows to design guaranteed, fully computable, constant-free, and polynomial-degree-robust a posteriori error estimates of Pra…
This paper constructs two conforming finite element grad grad and elasticity complexes on the cuboid meshes. For the finite element grad grad complex, an $H^2$ conforming finite element space, an $\boldsymbol{H}(\operatorname{curl}; \mathbb{S})$ conforming finite element space, an $\boldsymbol{H}(\operatorname{div}; \mathbb{T})$ conforming finite element space and an $\boldsymbol{L}^2$ finite element space are constructed. Further, a finite element complex with reduced regularity is also constructed, whose degrees of freedom for the three diagonal components are coupled. For the finite element…
An hp-version interior penalty discontinuous Galerkin (IPDG) method under nonconforming meshes is proposed to solve the quad-curl eigenvalue problem. We prove well-posedness of the numerical scheme for the quad-curl equation and then derive an error estimate in a mesh-dependent norm, which is optimal with respect to h but has different p-version error bounds under conforming and nonconforming tetrahedron meshes. The hp-version discrete compactness of the DG space is established for the convergence proof. The performance of the method is demonstrated by numerical experiments using conforming/no…
We investigate numerical solutions of high order curl problems with various formulations and finite elements. We show that several classical conforming finite elements lead to spurious solutions, while mixed formulations with finite elements in complexes solve the problems correctly. To explain the numerical results, we clarify the cohomological structures in high order curl problems by relating the partial differential equations to the Hodge-Laplacian boundary problems of the gradcurl-complexes.
Robust mixed finite element methods are developed for a quad-curl singular perturbation problem. Lower order H(grad curl)-nonconforming but H(curl)-conforming finite elements are constructed, which are extended to nonconforming finite element Stokes complexes and the associated commutative diagrams. Then H(grad curl)-nonconforming finite elements are employed to discretize the quad-curl singular perturbation problem, which possess the sharp and uniform error estimates with respect to the perturbation parameter. The Nitsche's technique is exploited to achieve the optimal convergence rate in the…
In this paper, we propose a new family of H(curl^2)-conforming elements for the quad-curl eigenvalue problem in 2D. The accuracy of this family is one order higher than that in [32]. We prove a priori and a posteriori error estimates. The a priori estimate of the eigenvalue with a convergence order 2(s-1) is obtained if the eigenvector u\in H^{s+1}(Ω). For the a posteriori estimate, by analyzing the associated source problem, we obtain lower and upper bounds for the eigenvector in an energy norm and an upper bound for the eigenvalues. Numerical examples are presented for validation.
We study extensions of piecewise polynomial data prescribed in a patch of tetrahedra sharing an edge. We show stability in the sense that the minimizers over piecewise polynomial spaces with prescribed tangential component jumps across faces and prescribed piecewise curl in elements are subordinate in the broken energy norm to the minimizers over the broken H(curl) space with the same prescriptions. Our proofs are constructive and yield constants independent of the polynomial degree. We then detail the application of this result to the a posteriori error analysis of the curl-curl problem discr…
Numerical analysis of a locking-free primal hybrid method for linear elasticity with $H(\mathrm{div})$-conforming stress recovery
Universidade Estadual de Campinas (Unicamp), Center for Energy and Petroleum Studies（坎皮纳斯州立大学（Unicamp），能源与石油研究中心）；Department of Applied Mathematics, Institute of Mathematics, Statistics and Computer Science, University of São Paulo（圣保罗大学，数学、统计与计算机科学研究所，应用数学系）
Two types of finite element spaces on a tetrahedron are constructed for divdiv conforming symmetric tensors in three dimensions. The key tools of the construction are the decomposition of polynomial tensor spaces and the characterization of the trace operators. First, the divdiv Hilbert complex and its corresponding polynomial complexes are presented. Several decompositions of polynomial vector and tensors spaces are derived from the polynomial complexes. Then, traces for div-div operator are characterized through a Green's identity. Besides the normal-normal component, another trace involving…
Finite element de Rham complexes and finite element Stokes complexes with various smoothness in three dimensions are systematically constructed. First smooth scalar finite elements in three dimensions are derived through a non-overlapping decomposition of the simplicial lattice. Based on the smooth scalar finite elements, both H(div)-conforming finite elements and H(curl)-conforming finite elements with various smoothness are devised, which induce the finite element de Rham complexes with various smoothness and the associated commutative diagrams. The div stability is established for the H(div…
Maxwell interface problems are of great importance in many electromagnetic applications. Unfitted mesh methods are especially attractive in 3D computation as they can circumvent generating complex 3D interface-fitted meshes. However, many unfitted mesh methods rely on non-conforming approximation spaces, which may cause a loss of accuracy for solving Maxwell equations, and the widely-used penalty techniques in the literature may not help in recovering the optimal convergence. In this article, we provide a remedy by developing Nédélec-type immersed finite element spaces with a Petrov-Galerkin s…
H(curl)-based approximation of the Stokes problem with slip boundary conditions
Faculty of Mathematics, University of Duisburg-Essen（杜伊斯堡-埃森大学数学学院）；SAM, ETH Zürich（苏黎世联邦理工学院应用数学系）；Department of Mathematics, University of Vienna（维也纳大学数学系）
We address fundamental aspects in the approximation theory of vector-valued finite element methods, using finite element exterior calculus as a unifying framework. We generalize the Clément interpolant and the Scott-Zhang interpolant to finite element differential forms, and we derive a broken Bramble-Hilbert Lemma. Our interpolants require only minimal smoothness assumptions and respect partial boundary conditions. This permits us to state local error estimates in terms of the mesh size. Our theoretical results apply to curl-conforming and divergence-conforming finite element methods over sim…
We derive a posteriori error estimates for a semi-discrete finite element approximation of a nonlinear eddy current problem arising from applied superconductivity, known as the $p$-curl problem. In particular, we show the reliability for non-conforming Nédélec elements based on a residual type argument and a Helmholtz-Weyl decomposition of $W^p_0(\text{curl};Ω)$. As a consequence, we are also able to derive an a posteriori error estimate for a quantity of interest called the AC loss. The nonlinearity for this form of Maxwell's equation is an analogue of the one found in the $p$-Laplacian. It i…
Two nonconforming finite element Stokes complexes starting from the conforming Lagrange element and ending with the nonconforming $P_1$-$P_0$ element for the Stokes equation in three dimensions are constructed. And commutative diagrams are also shown by combining nonconforming finite element Stokes complexes and interpolation operators. The lower order $\boldsymbol H(\textrm{grad}\textrm{curl})$-nonconforming finite element only has $14$ degrees of freedom, whose basis functions are explicitly given in terms of the barycentric coordinates. The $\boldsymbol H(\textrm{grad}\textrm{curl})$-noncon…
Given a sequence of finite element spaces which form a de Rham sequence, we will construct a dual representation of these spaces with associated differential operators which connect these spaces such that they also form a de Rham sequence. The matrix which converts primal representations to dual representations -- the Hodge matrix -- is the mass or Gram matrix. It will be shown that a bilinear form of a primal and a dual representation is equal to the vector inner product of the expansion coefficients (degrees of freedom) of both representations. This leads to very sparse system matrices, even…
Wave propagation problems are notoriously difficult to solve. Time-harmonic problems are especially challenging in mid and high frequency regimes. The main reason is the oscillatory nature of solutions, meaning that the number of degrees of freedom after discretisation increases drastically with the wave number, giving rise to large complex-valued problems to solve. Additional difficulties occur when the problem is defined in a highly heterogeneous medium, as is often the case in realistic physical applications. For time-discretised problems of Maxwell type, the main challenge remains the sign…
The Hilbert spaces $H(\mathrm{curl})$ and $H(\mathrm{div})$ are needed for variational problems formulated in the context of the de Rham complex in order to guarantee well-posedness. Consequently, the construction of conforming subspaces is a crucial step in the formulation of viable numerical solutions. Alternatively to the standard definition of a finite element as per Ciarlet, given by the triplet of a domain, a polynomial space and degrees of freedom, this work aims to introduce a novel, simple method of directly constructing semi-continuous vectorial base functions on the reference elemen…
