Related papers: Numerical solution of the div-curl problem by fini…
We consider a semi-discrete finite element approximation for a system consisting of the evolution of a planar curve evolving by forced curve shortening flow inside a given bounded domain $\Omega \subset \mathbb{R}^2$, such that the curve…
Discrete exterior calculus (DEC) is a structure-preserving numerical framework for partial differential equations solution, particularly suitable for simplicial meshes. A longstanding and widespread assumption has been that DEC requires…
In this work we study three exterior extension problems for strongly elliptic partial equations: the Cauchy problem (in a special statement), the "analytical" continuation problem and the so called "inner" Dirichlet problem in the scale of…
We develop finite element exterior calculus over weakly Lipschitz domains. Specifically, we construct commuting projections from $L^p$ de~Rham complexes over weakly Lipschitz domains onto finite element de~Rham complexes. These projections…
Many important physical problems, such as fluid structure interaction or conjugate heat transfer, require numerical methods that compute boundary derivatives or fluxes to high accuracy. This paper proposes a novel alternative to calculating…
This survey revisits classical results in vector calculus and analysis by exploring a generalised perspective on the exterior derivative, interpreting it as a measure of "infinitesimal flux". This viewpoint leads to a higher-dimensional…
In this paper, we study a semilinear system involving the curl operator in a bounded and convex domain in $R^3$, which comes from the steady-state approximation for Bean critical-state model for type-II superconductors. We show the…
We numerically implement the variational approach for reconstruction in the inverse crack and cavity problems developed by one of the authors. The method is based on a suitably adapted free-discontinuity problem. Its main features are the…
An arbitrary order finite difference method for curved boundary domains with Cartesian grid is proposed. The technique handles in a universal manner Dirichlet, Neumann or Robin condition. We introduce the Reconstruction Off-site Data (ROD)…
We present a new finite element method, called $\phi$-FEM, to solve numerically elliptic partial differential equations with natural (Neumann or Robin) boundary conditions using simple computational grids, not fitted to the boundary of the…
Fractional vector calculus is discussed in the spherical coordinate framework. A variation of the Legendre equation and fractional Bessel equation are solved by series expansion and numerically. Finally, we generalize the hypergeometric…
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…
Using continuation methods, we study the global solution structure of periodic solutions for a class of periodically forced equations, generalizing the case of relativistic pendulum. We obtain results on the existence and multiplicity of…
This paper deals with bounding the error on the estimation of quantities of interest obtained by finite element and domain decomposition methods. The proposed bounds are written in order to separate the two errors involved in the resolution…
Motivated by many applications in complex domains with boundaries exposed to large topological changes or deformations, fictitious domain methods regard the actual domain of interest as being embedded in a fixed Cartesian background. This…
We construct local projections into canonical finite element spaces that appear in the finite element exterior calculus. These projections are bounded in $L^2$ and commute with the exterior derivative.
This paper proposes direct and inverse results for the Dirichlet and Dirichlet to Neumann problems for complex curves with nodal type singularities. As an application, we give a method to reconstruct the conformal structure of a compact…
In the present paper we construct Virtual Element Spaces that are $H({\rm div})$-conforming and $H({\rm \bf curl})$-conforming on general polygonal and polyhedral elements; these spaces can be interpreted as a generalization of well known…
There are many numerical methods for solving partial different equations (PDEs) on manifolds such as classical implicit, finite difference, finite element, and isogeometric analysis methods which aim at improving the interoperability…
Using the method of covariant symbols we compute the divergent part of the effective action of the Proca field with non-minimal mass term. Specifically a quantum Abelian vector field with a non-derivative coupling to an external tensor…