Related papers: Finite Element Complexes in Two Dimensions
As inelastic structures are ubiquitous in many engineering fields, a central task in computational mechanics is to develop accurate, robust and efficient tools for their analysis. Motivated by the poor performances exhibited by standard…
We construct finite element de~Rham complexes of higher and possibly non-uniform polynomial order in finite element exterior calculus (FEEC). Starting from the finite element differential complex of lowest-order, known as the complex of…
We propose a two-scale finite element method designed for heterogeneous microstructures. Our approach exploits domain diffeomorphisms between the microscopic structures to gain computational efficiency. By using a conveniently constructed…
The purpose of this paper is to discuss a generalization of the bubble transform to differential forms. The bubble transform was discussed in a previous paper by the authors for scalar valued functions, or zero-forms, and represents a new…
We design in this work a discrete de Rham complex on manifolds. This complex, written in the framework of exterior calculus, has the same cohomology as the continuous de Rham complex, is of arbitrary order of accuracy and, in principle, can…
We present in this paper a rigorous theoretical framework to show stability, convergence and accuracy of improved edge-based and face-based smoothed finite element methods (bESFEM and bFS-FEM) for nearly-incompressible elasticity problems.…
We develop a method to compute $H^2$-conforming finite element approximations in both two and three space dimensions using readily available finite element spaces. This is accomplished by deriving a novel, equivalent mixed variational…
The tensor product of two differential forms of degree $p$ and $q$ is a multilinear form that is alternating in its first $p$ arguments and alternating in its last $q$ arguments. These forms, which are known as double forms or…
We reconsider non-degenerate second order superintegrable systems in dimension two as geometric structures on conformal surfaces. This extends a formalism developed by the authors, initially introduced for (pseudo-)Riemannian manifolds of…
We give a systematic and self-contained account of the construction of geometrically decomposed bases and degrees of freedom in finite element exterior calculus. In particular, we elaborate upon a previously overlooked basis for one of the…
This article reports on the confluence of two streams of research, one emanating from the fields of numerical analysis and scientific computation, the other from topology and geometry. In it we consider the numerical discretization of…
We present a class of discretisation spaces and H(div)-conformal elements that can be built on any polytope. Bridging the flexibility of the Virtual Element spaces towards the element's shape with the divergence properties of the…
This paper introduces a novel tangential-normal ($t$-$n$) decomposition for finite element differential forms, presenting a new framework for constructing bases in finite element exterior calculus. The main contribution is the development…
Accurately depicting multiphysics interactions in interfacial systems requires computational frameworks capable of reconciling geometric adaptability with strict conservation fidelity. However, traditional spatiotemporal discretisation…
In this work, following the discrete de Rham (DDR) approach, we develop a discrete counterpart of a two-dimensional de Rham complex with enhanced regularity. The proposed construction supports general polygonal meshes and arbitrary…
We present a multiscale mixed finite element method for solving second order elliptic equations with general $L^{\infty}$-coefficients arising from flow in highly heterogeneous porous media. Our approach is based on a multiscale spectral…
Using the language of finite element exterior calculus, we define two families of $H^1$-conforming finite element spaces over pyramids with a parallelogram base. The first family has matching polynomial traces with tensor product elements…
In this paper, gradient-based optimization methods are combined with finite-element modeling for improving electric devices. Geometric design parameters are considered by affine decomposition of the geometry or by the design element…
We present degrees of freedom to accompany the approximation spaces already presented in a companion paper and thus complete the definition of families of high-order conforming finite elements on pyramids for the spaces of the de Rham…
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…