Related papers: Numerical solution of the div-curl problem by fini…
Fractional vector calculus is the building block of the fractional partial differential equations that model non-local or long-range phenomena, e.g., anomalous diffusion, fractional electromagnetism, and fractional advection-dispersion. In…
In the paper we study the 2D div-curl problem in the exterior domain which models the flow with given vorticity, divergency, boundary condition at infinity, and Dirichlet condition on the solid surface. We will find the relations on…
A generalization of exterior calculus is considered by allowing the partial derivatives in the exterior derivative to assume fractional orders. That is, a fractional exterior derivative is defined. This is found to generate new vector…
The paper addresses the challenge of constructing finite element curl div complexes in three dimensions. Tangential-normal continuity is introduced in order to develop distributional finite element curl div complexes. The spaces constructed…
We present a theory and applications of discrete exterior calculus on simplicial complexes of arbitrary finite dimension. This can be thought of as calculus on a discrete space. Our theory includes not only discrete differential forms but…
We consider the inhomogeneous div-curl system (i.e.\ to find a vector field with prescribed div and curl) in a bounded star-shaped domain in $\mathbb{R}^3$. An explicit general solution is given in terms of classical integral operators,…
This article addresses the solvability of the multi-dimensional divergence-curl problem with a no-slip boundary condition. A solvability criterion is derived as an orthogonality condition of the vorticity function to pseudo-harmonic fields.…
Finite element approximation to a decoupled formulation for the quad--curl problem is studied in this paper. The difficulty of constructing elements with certain conformity to the quad--curl problems has been greatly reduced. For convex…
We design and analyze virtual element methods for a quad-curl problem on general polygonal domains that are based on the Hodge decomposition of divergence-free vector fields. Numerical results that corroborate the theoretical analysis are…
We prove a global version of the so-called div-curl-lemma, a crucial result for compensated compactness and in homogenization theory, for mixed tangential and normal boundary conditions in bounded weak Lipschitz domains in 3D and weak…
We devise and analyze hybrid polyhedral methods of arbitrary order for the approximation of div-curl systems on three-dimensional domains featuring non-trivial topology. The div-curl systems we are interested in stem from magnetostatics,…
In this paper, we perform a numerical analysis in frequency domain for various electromagnetic problems based on discrete exterior calculus (DEC) with an arbitrary 2-D triangular or 3-D tetrahedral mesh. We formulate the governing equations…
We prove global and local versions of the so-called div-curl-lemma, a crucial result in the homogenization theory of partial differential equations, for mixed boundary conditions on bounded weak Lipschitz domains in 3D with weak Lipschitz…
A construction of prismatic Hardy space infinite elements to discretize wave equations on unbounded domains $\Omega$ in $H^1_{loc}(\Omega)$, $H_{loc}(curl;\Omega)$ and $H_{loc}(div;\Omega)$ is presented. As our motivation is to solve…
The problem of solving partial differential equations (PDEs) on manifolds can be considered to be one of the most general problem formulations encountered in computational multi-physics. The required covariant forms of balance laws as well…
Over the last ten years, the Finite Element Exterior Calculus (FEEC) has been developed as a general framework for linear mixed variational problems, their numerical approximation by mixed methods, and their error analysis. The basic…
Simplicial, piecewise-flat discretizations of manifolds provide a clear path towards curvature analysis on discrete geometries and for solutions of PDE's on manifolds of complex topologies. In this manuscript we review and expand on…
In this paper we present a nonconforming finite element method for solving fourth order curl equations in three dimensions arising from magnetohydrodynamics models. We show that the method has an optimal error estimate for a model problem…
We survey recent contributions to finite element exterior calculus on manifolds and surfaces within a comprehensive formalism for the error analysis of vector-valued partial differential equations on manifolds. Our primary focus is on…
We present a family of nonconforming vector finite elements of arbitrary order for problems posed on the space (curl) intersected with H(div) on a bidimensional domain. This result was first stated as a conjecture by Brenner and Sung. In…