Related papers: Optimal Meshes for Finite Elements of Arbitrary Or…
In recent years, a number of finite element methods have been formulated for the solution of partial differential equations on complex geometries based on non-matching or overlapping meshes. Examples of such methods include the fictitious…
We propose a new algorithm for the design of topologically optimized lightweight structures, under a minimum compliance requirement. The new process enhances a standard level set formulation in terms of computational efficiency, thanks to…
For the discretisation of $H_{div}$-functions on rectangular meshes there are at least three families of elements, namely Raviart-Thomas-, Brezzi-Douglas-Marini- and Arnold-Boffi-Falk-elements. In order to prove convergence of a numerical…
The convergence and optimality of adaptive mixed finite element methods for the Poisson equation are established in this paper. The main difficulty for mixed finite element methods is the lack of minimization principle and thus the failure…
This is a study of certain finite element methods designed for convection-dominated, time-dependent partial differential equations. Specifically, we analyze high order space-time tensor product finite element discretizations, used in a…
Finite element approximations of minimal surface are not always precise. They can even sometimes completely collapse. In this paper, we provide a simple and inexpensive method, in terms of computational cost, to improve finite element…
We introduce a novel type of approximation spaces for functions with values in a nonlinear manifold. The discrete functions are constructed by piecewise polynomial interpolation in a Euclidean embedding space, and then projecting pointwise…
In the context of unfitted finite element discretizations the realization of high order methods is challenging due to the fact that the geometry approximation has to be sufficiently accurate. Recently a new unfitted finite element method…
The paper shows an inf-sup stability property for several well-known 2D and 3D Stokes elements on triangulations which are not fitted to a given smooth or polygonal domain. The property implies stability and optimal error estimates for a…
In [Heimann, Lehrenfeld, Preu{\ss}, SIAM J. Sci. Comp. 45(2), 2023, B139 - B165] new geometrically unfitted space-time Finite Element methods for partial differential equations posed on moving domains of higher-order accuracy in space and…
We consider shape optimization problems subject to elliptic partial differential equations. In the context of the finite element method, the geometry to be optimized is represented by the computational mesh, and the optimization proceeds by…
This paper studies adaptive first-order least-squares finite element methods for second-order elliptic partial differential equations in non-divergence form. Unlike the classical finite element method which uses weak formulations of PDEs…
In this paper, we propose a new approach -- the Tempered Finite Element Method (TFEM) -- that extends the Finite Element Method (FEM) to classes of meshes that include zero-measure or nearly degenerate elements for which standard FEM…
We design an adaptive finite element method to approximate the solutions of quasi-linear elliptic problems. The algorithm is based on a Ka\v{c}anov iteration and a mesh adaptation step is performed after each linear solve. The method is…
Some error analysis on virtual element methods including inverse inequalities, norm equivalence, and interpolation error estimates are presented for polygonal meshes which admits a virtual quasi-uniform triangulation.
We prove optimal order error estimates for the Raviart-Thomas interpolation of arbitrary order under the maximum angle condition for triangles and under two generalizations of this condition, namely, the so-called three dimensional maximum…
Mesh adaptivity is a useful tool for efficient solution to partial differential equations in very complex geometries. In the present paper we discuss the use of polygonal mesh refinement in order to tackle two common issues: first,…
We consider scalar semilinear elliptic PDEs where the nonlinearity is strongly monotone, but only locally Lipschitz continuous. We formulate an adaptive iterative linearized finite element method (AILFEM) which steers the local mesh…
The a posteriori error analysis of the classical Argyris finite element methods dates back to 1996, while the optimal convergence rates of associated adaptive finite element schemes are established only very recently in 2021. It took a long…
In \cite{liu2022practical}, a general algorithm is developed to efficiently obtain the best accuracy using the regular refinement. The adaptive refinement allows for obtaining an accuracy with a smaller number of DoFs compared with the…