Related papers: Finite Element Complexes in Two Dimensions
This work deals with planar dynamical systems with and without noise. In the first part, we seek to gain a refined understanding of such systems by studying their differential-geometric transformation properties under an arbitrary smooth…
Recovered finite element methods (R-FEM) have been recently introduced for meshes consisting of simplicial and/or box-type meshes. Here, utilising the flexibility of R-FEM framework, we extend their definition on polygonal and polyhedral…
This paper is concerned with the derivation and properties of differential complexes arising from a variety of problems in differential equations, with applications in continuum mechanics, relativity, and other fields. We present a…
In this thesis we examine a set of foundational questions concerning closed forms in superspace. By reformulating a number of definitions through the use of a new ring of (anti-)commuting variables and the concept of an exact Bianchi form,…
This paper presents an innovative continuous linear finite element approach to effectively solve biharmonic problems on surfaces. The key idea behind this method lies in the strategic utilization of a surface gradient recovery operator to…
Upper bound limit analysis allows one to evaluate directly the ultimate load of structures without performing a cumbersome incremental analysis. In order to numerically apply this method to thin plates in bending, several authors have…
The biharmonic equation with Dirichlet and Neumann boundary conditions discretized using the mixed finite element method and piecewise linear (with the possible exception of boundary triangles) finite elements on triangular elements has…
In this paper we present a novel arbitrary-order discrete de Rham (DDR) complex on general polyhedral meshes based on the decomposition of polynomial spaces into ranges of vector calculus operators and complements linked to the spaces in…
This paper studies the discretization of a homogenization and dimension reduction model for the elastic deformation of microstructured thin plates proposed by Hornung, Neukamm, and Vel\v{c}i\'c in 2014. Thereby, a nonlinear bending energy…
Fitted finite element methods are constructed for a singularly perturbed convection-diffusion problem in two space dimensions. Exponential splines as basis functions are combined with Shishkin meshes to obtain a stable parameter-uniform…
We introduce a pressure robust Finite Element Method for the linearized Magnetohydrodynamics equations in three space dimensions, which is provably quasi-robust also in the presence of high fluid and magnetic Reynolds numbers. The proposed…
The classical serendipity and mixed finite element spaces suffer from poor approximation on nondegenerate, convex quadrilaterals. In this paper, we develop $\textit{direct serendipity}$ and $\textit{direct mixed}$ finite element spaces,…
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…
Let A be a commutative ring, B a commutative A-algebra and M a complex of B-modules. We begin by constructing the square Sq_{B/A} M, which is also a complex of B-modules. The squaring operation is a quadratic functor, and its construction…
A family of mixed finite elements is proposed for solving the first order system of linear elasticity equations in any space dimension, where the stress field is approximated by symmetric finite element tensors. This family of elements has…
For the finite element simulation of thin soft biological tissues in dynamics, shell elements, compared to volume elements, can capture the whole tissue thickness at once, and feature larger critical time steps. However, the capabilities of…
This paper introduces discrete differential form spaces over two-dimensional manifold meshes that feature enhanced subdivision-induced inter-element regularity compared to conventional finite element (FE) spaces. This increase in smoothness…
Computing the stiffness matrix for the finite element discretization of the nonlocal Laplacian on unstructured meshes is difficult, because the operator is nonlocal and can even be singular. In this paper, we focus on the $C^0$-piecewise…
This paper develops a complete foundational treatment of simplicial complexes from Euclidean spaces through geometric realizations, emphasizing concrete computations, examples, and practical verification methods. Beginning with finite point…
The quantum group version of the Bernstein-Gelfand-Gelfand resolution is used to construct a double complex of U_q(g)-modules with exact rows and columns. The locally finite dual of its total complex is identified with the de Rham complex…