More Like this

1. Residual-based a posteriori error estimation for hp-adaptive finite element methods for the stokes equations

   https://doi.org/10.1515/jnma-2018-0047  

   Ghesmati, A.; Bangerth, W.; Turcksin, B. (November 2018, Journal of Numerical Mathematics)

   Abstract We derive a residual-based a posteriori error estimator for the conforming hp -Adaptive Finite Element Method ( hp -AFEM) for the steady state Stokes problem describing the slow motion of an incompressible fluid. This error estimator is obtained by extending the idea of a posteriori error estimation for the classical h -version of AFEM. We also establish the reliability and efficiency of the error estimator. The proofs are based on the well-known Clément-type interpolation operator introduced in [28] in the context of the hp -AFEM. Numerical experiments show the performance of an adaptive hp-FEM algorithm using the proposed a posteriori error estimator. 

   more » « less
2. Convergence of an adaptive finite element DtN method for the elastic wave scattering by periodic structures

   https://doi.org/10.1016/j.cma.2019.112722  

   Li, P.; Yuan, X. (January 2020, Computer methods in applied mechanics and engineering)

   Consider the scattering of a time-harmonic elastic plane wave by a periodic rigid surface. The elastic wave propagation is governed by the two-dimensional Navier equation. Based on a Dirichlet-to-Neumann (DtN) map, a transparent boundary condition (TBC) is introduced to reduce the scattering problem into a boundary value problem in a bounded domain. By using the finite element method, the discrete problem is considered, where the TBC is replaced by the truncated DtN map. A new duality argument is developed to derive the a posteriori error estimate, which contains both the finite element approximation error and the DtN truncation error. An a posteriori error estimate based adaptive finite element algorithm is developed to solve the elastic surface scattering problem. Numerical experiments are presented to demonstrate the effectiveness of the proposed method. 

   more » « less
3. A finite element method for a two-dimensional Pucci equation

   https://doi.org/10.5802/crmeca.224  

   Brenner, Susanne C; Sung, Li-yeng; Tan, Zhiyu (April 2024, Comptes Rendus. Mécanique)

   A nonlinear least-squares finite element method for strong solutions of the Dirichlet boundary value problem of a two-dimensional Pucci equation on convex polygonal domains is investigated in this paper. We obtain a priori and a posteriori error estimates and present corroborating numerical results, where the discrete nonsmooth and nonlinear optimization problems are solved by an active set method and an alternating direction method with multipliers. 

   more » « less
4. A convexity enforcing C0 interior penalty method for the Monge–Ampère equation on convex polygonal domains

   Brenner, Susanne C; Sung, Li-yeng; Tan, Zhiyu; Zhang, Hongchao (June 2021, Numerische Mathematik)

   We design and analyze a C0 interior penalty method for the approximation of classical solutions of the Dirichlet boundary value problem of the Monge–Ampère equation on convex polygonal domains. The method is based on an enhanced cubic Lagrange finite element that enables the enforcement of the convexity of the approximate solutions. Numerical results that corroborate the a priori and a posteriori error estimates are presented. It is also observed from numerical experiments that this method can capture certain weak solutions. 

   more » « less
5. Quasi-optimality of a pressure-robust nonconforming finite element method for the Stokes-Problem

   https://doi.org/10.1090/mcom/3344  

   A. Linke, C. Merdon (January 2018, Mathematics of computation)

   Nearly all classical inf-sup stable mixed finite element methods for the incompressible Stokes equations are not pressure-robust, i.e., the velocity error is dependent on the pressure. However, recent results show that pressure-robustness can be recovered by a nonstandard discretization of the right-hand side alone. This variational crime introduces a consistency error in the method which can be estimated in a straightforward manner provided that the exact velocity solution is sufficiently smooth. The purpose of this paper is to analyze the pressure-robust scheme with low regularity. The numerical analysis applies divergence-free $H^1$-conforming Stokes finite element methods as a theoretical tool. As an example, pressure-robust velocity and pressure a priori error estimates will be presented for the (first-order) nonconforming Crouzeix-Raviart element. A key feature in the analysis is the dependence of the errors on the Helmholtz projector of the right-hand side data, and not on the entire data term. Numerical examples illustrate the theoretical results. 

   more » « less