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A finite element methodology for large classes of variational boundary value problems is defined which involves discretizing two linear operators: (1) the differential operator defining the spatial boundary value problem; and (2) a Riesz…

Numerical Analysis · Mathematics 2017-12-08 Brendan Keith , Socratis Petrides , Federico Fuentes , Leszek Demkowicz

We consider a new fictitious domain approach of higher order accuracy. To implement Dirichlet conditions we apply the classical Nitsche method combined with a facet-based stabilization (ghost penalty). Both techniques are combined with a…

Numerical Analysis · Mathematics 2017-07-04 Christoph Lehrenfeld

We propose a framework for unified analysis of mixed methods for elasticity with weakly symmetric stress. Based on a commuting diagram in the weakly symmetric elasticity complex and extending a previous stability result, stable mixed…

Numerical Analysis · Mathematics 2015-10-12 Jeonghun J. Lee

This paper presents a method for the optimization of multi-component structures comprised of two and three materials considering large motion sliding contact and separation along interfaces. The structural geometry is defined by an explicit…

Optimization and Control · Mathematics 2017-01-24 Matthew Lawry , Kurt Maute

A stable numerical solution of the steady Stokes problem requires compatibility between the choice of velocity and pressure approximation that has traditionally proven problematic for meshless methods. In this work, we present a…

Numerical Analysis · Mathematics 2016-11-15 Nathaniel Trask , Martin Maxey , Xiaozhe Hu

In this paper, we present a novel local and parallel two-grid finite element scheme for solving the Stokes equations, and rigorously establish its a priori error estimates. The scheme admits simultaneously small scales of subproblems and…

Numerical Analysis · Mathematics 2020-12-09 Yanren Hou , Feng Shi , Haibiao Zheng

We introduce and analyze a robust nonconforming finite element method for a three dimensional singularly perturbed quad-curl model problem. For the solution of the model problem, we derive proper a priori bounds, based on which, we prove…

Numerical Analysis · Mathematics 2022-06-27 Baiju Zhang , Zhimin Zhang

We extend the divergence preserving cut finite element method presented in [T. Frachon, P. Hansbo, E. Nilsson, S. Zahedi, SIAM J. Sci. Comput., 46 (2024)] for the Darcy interface problem to unfitted outer boundaries. We impose essential…

Numerical Analysis · Mathematics 2024-08-20 Thomas Frachon , Erik Nilsson , Sara Zahedi

We present the first convergence proof for an iso-parametric finite element discretization of two-phase Stokes flow in $\Omega \subset \mathbb{R}^d$, $d=2,3$, with interface dynamics governed by mean curvature. The proof relies on a crucial…

Numerical Analysis · Mathematics 2025-09-25 Genming Bai , Harald Garcke , Shravan Veerapaneni

We introduce a new discretization of a mixed formulation of the incompressible Stokes equations that includes symmetric viscous stresses. The method is built upon a mass conserving mixed formulation that we recently studied. The improvement…

Numerical Analysis · Mathematics 2024-12-20 Jay Gopalakrishnan , Philip L. Lederer , Joachim Schöberl

A series of robust and optimal mixed methods based on two mixed formulations of the fourth-order elliptic singular perturbation problem are developed in this paper. First, a mixed method based on a second-order system is proposed without…

Numerical Analysis · Mathematics 2025-09-18 Xuehai Huang , Zheqian Tang

A parallel implementation of a compatible discretization scheme for steady-state Stokes problems is presented in this work. The scheme uses generalized moving least squares to generate differential operators and apply boundary conditions.…

Numerical Analysis · Mathematics 2021-04-30 Quang-Thinh Ha , Paul A. Kuberry , Nathaniel A. Trask , Emily M. Ryan

This work aims to solve a stochastic nonconvex nonsmooth composite optimization problem. Previous works on composite optimization problem requires the major part to satisfy Lipschitz smoothness or some relaxed smoothness conditions, which…

Optimization and Control · Mathematics 2025-10-07 Ziyi Chen , Peiran Yu , Heng Huang

We consider solving the surface Helmholtz equation on a smooth two dimensional surface embedded into a three dimensional space meshed with tetrahedra. The mesh does not respect the surface and thus the surface cuts through the elements. We…

Numerical Analysis · Mathematics 2018-10-11 Erik Burman , Peter Hansbo , Mats G. Larson , Andre Massing

A Nitche's method is presented to couple non-conforming two and three dimensional NURBS (Non Uniform Rational B-splines) patches in the context of isogeometric analysis (IGA). We present results for elastic stress analyses under the static…

Numerical Analysis · Mathematics 2016-02-01 Vinh Phu Nguyen , Pierre Kerfriden , Marco Brino , Stephane Bordas , Elvio Bonisoli

We present a numerical approximation method for linear diffusion-reaction problems with possibly discontinuous Dirichlet boundary conditions. The solution of such problems can be represented as a linear combination of explicitly known…

Numerical Analysis · Mathematics 2017-07-05 Ramona Baumann , Thomas P. Wihler

We introduce an arbitrary order, stabilized finite element method for solving a unique continuation problem subject to the time-harmonic elastic wave equation with variable coefficients. Based on conditional stability estimates we prove…

Numerical Analysis · Mathematics 2023-04-25 Erik Burman , Janosch Preuss

A two-grid scheme based on mixed finite-element approximations to the incompressible Navier-Stokes equations is introduced and analyzed. In the first level the standard mixed finite-element approximation over a coarse mesh is computed. In…

Numerical Analysis · Mathematics 2016-12-23 Javier de Frutos , Bosco García-Archilla , Julia Novo

We present and analyze a cut finite element method for the weak imposition of the Neumann boundary conditions of the Darcy problem. The Raviart-Thomas mixed element on both triangular and quadrilateral meshes is considered. Our method is…

Numerical Analysis · Mathematics 2021-11-22 Riccardo Puppi

We study the Stokes problem in a bounded planar domain $\Omega$ with a friction type boundary condition that switches between a slip and no-slip stage. Unlike our previous work [6], in the present paper the threshold value may depend on the…

Analysis of PDEs · Mathematics 2016-01-22 Jaroslav Haslinger , Jan Stebel