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1. A posteriori error estimates of Darcy flows with Robin-type jump interface conditions

   Lee, Jeonghun J (November 2024, Computers and mathematics with applications)

   In this work we develop an a posteriori error estimator for mixed finite element methods of Darcy flow problems with Robin-type jump interface conditions. We construct an energy-norm type a posteriori error estimator using the Stenberg post-processing. The reliability of the estimator is proved using an interface-adapted Helmholtz-type decomposition and an interface-adapted Scott-Zhang type interpolation operator. A local efficiency and the reliability of post-processed pressure are also proved. Numerical results illustrating adaptivity algorithms using our estimator are included. 

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2. 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. 

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3. 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. 

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4. 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. 

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5. 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. 

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