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A refined a priori error analysis of the lowest order (linear) nonconforming Virtual Element Method (VEM) for approximating a model Poisson problem is developed in both 2D and 3D. A set of new geometric assumptions is proposed on shape…

Numerical Analysis · Mathematics 2019-05-17 Shuhao Cao , Long Chen

We consider a vector-Laplace problem posed on a 2D surface embedded in a 3D domain, which results from the modeling of surface fluids based on exterior Cartesian differential operators. The main topic of this paper is the development and…

Numerical Analysis · Mathematics 2017-09-05 Sven Groß , Thomas Jankuhn , Maxim A. Olshanskii , Arnold Reusken

We propose and analyze an augmented mixed finite element method for the pseudostress-velocity formulation of the stationary convective Brinkman-Forchheimer problem in $\mathrm{R}^d$, $d\in \{2,3\}$. Since the convective and Forchheimer…

Numerical Analysis · Mathematics 2023-03-03 Sergio Caucao , Johann Esparza

We consider a mathematical model which describes the quasistatic frictionless contact of a viscoelastic body with a rigid-plastic foundation. We describe the mechanical assumptions, list the hypotheses on the data and provide three…

Analysis of PDEs · Mathematics 2023-09-11 Piotr Bartman , Anna Ochal , Mircea Sofonea

In the spirit of the "Principle of Equipresence" introduced by Truesdell & Toupin, The Classical Field Theories (1960), we use the full version of the viscous stress tensor which was originally derived for compressible flows, instead of the…

Numerical Analysis · Mathematics 2020-07-17 Xi Chen , David M. Williams

We consider a strongly heterogeneous medium saturated by an incompressible viscous fluid as it appears in geomechanical modeling. This poroelasticity problem suffers from rapidly oscillating material parameters, which calls for a thorough…

Numerical Analysis · Mathematics 2018-12-27 Robert Altmann , Eric Chung , Roland Maier , Daniel Peterseim , Sai-Mang Pun

In the present article, we are interested in an initial boundary value problem for a coupled system of partial differential equations arising in martensitic phase transition theory of elastically deformable solid materials, e.g., steel.…

Dynamical Systems · Mathematics 2011-02-07 Peicheng Zhu

We present a stabilized finite element method for the numerical solution of cavitation in lubrication, modeled as an inequality-constrained Reynolds equation. The cavitation model is written as a variable coefficient saddle-point problem…

Numerical Analysis · Mathematics 2018-05-09 Tom Gustafsson , K. R. Rajagopal , Rolf Stenberg , Juha Videman

This paper aims first to perform robust continuous analysis of a mixed nonlinear formulation for stress-assisted diffusion of a solute that interacts with an elastic material, and second to propose and analyse a virtual element formulation…

Numerical Analysis · Mathematics 2025-10-23 Rekha Khot , Andres E. Rubiano , Ricardo Ruiz-Baier

The virtual element method (VEM) allows discretization of elasticity and plasticity problems with polygons in 2D and polyhedrals in 3D. The polygons (and polyhedrals) can have an arbitrary number of sides and can be concave or convex. These…

Numerical Analysis · Mathematics 2023-12-05 Louie L. Yaw

This paper constructs the first mixed finite element for the linear elasticity problem in 3D using $P_3$ polynomials for the stress and discontinuous $P_2$ polynomials for the displacement on tetrahedral meshes under some mild mesh…

Numerical Analysis · Mathematics 2023-08-22 Jun Hu , Rui Ma , Yuanxun Sun

We introduce a stabilised finite element formulation for the Kirchhoff plate obstacle problem and derive both a priori and residual-based a posteriori error estimates using conforming $C^1$-continuous finite elements. We implement the…

Numerical Analysis · Mathematics 2021-02-09 Tom Gustafsson , Rolf Stenberg , Juha Videman

The high Weissenberg number problem has been a persistent challenge in the numerical simulation of viscoelastic fluid flows. This paper presents an improved lattice Boltzmann method for solving viscoelastic flow problems at high Weissenberg…

Fluid Dynamics · Physics 2025-08-26 Yuan Yu , Siwei Chen , Lei Wang , Hai-zhuan Yuan , Shi Shu

We study a model for the deformation of a visco-elasto-plastic material that is nearly incompressible. It originates from geophysics, is given in the Eulerian description and combines a Kelvin-Voigt rheology in the spherical part with a…

Analysis of PDEs · Mathematics 2025-12-22 Thomas Eiter

Hybrid kinetic-MHD models describe the interaction of an MHD bulk fluid with an ensemble of hot particles, which is described by a kinetic equation. When the Vlasov description is adopted for the energetic particles, different Vlasov-MHD…

Plasma Physics · Physics 2017-03-21 Joshua W. Burby , Cesare Tronci

We consider the linear system of viscoelasticity with the homogeneous Dirichlet boundary condition. First we prove a Carleman estimate with boundary values of solutions of viscoelasticity system. Since a solution $u$ under consideration is…

Analysis of PDEs · Mathematics 2017-11-28 Oleg Imanuvilov , Masahiro Yamamoto

We consider the applications of a numerical-analytical approach based on multiscale variational wavelet technique to the systems with collective type behaviour described by some forms of Vlasov-Poisson/Maxwell equations. We calculate the…

Accelerator Physics · Physics 2007-05-23 Antonina N. Fedorova , Michael G. Zeitlin

This paper presents a modified general viscosity iterative process designed to solve variational inclusion and fixed point problems involving multi-valued quasi-nonexpansive and demi-contractive operators. The modified iterative process…

Optimization and Control · Mathematics 2025-01-14 Furmose Mendy , John T Mendy

An extended volume of fluid method is developed for two-phase direct numerical simulations of systems with one viscoelastic and one Newtonian phase. A complete set of governing equations is derived by conditional volume-averaging of the…

Fluid Dynamics · Physics 2020-12-08 Matthias Niethammer , Günter Brenn , Holger Marschall , Dieter Bothe

In this paper, we introduce new stable mixed finite elements of any order on polytopal mesh for solving second order elliptic problem. We establish optimal order error estimates for velocity and super convergence for pressure. Numerical…

Numerical Analysis · Mathematics 2020-09-14 Yanping Lin , Xiu Ye , Shangyou Zhang