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1. An Eulerian finite element method for the linearized Navier–Stokes problem in an evolving domain

   https://doi.org/10.1093/imanum/drad105  

   Neilan, Michael; Olshanskii, Maxim (January 2024, IMA Journal of Numerical Analysis)

   Abstract The paper addresses an error analysis of an Eulerian finite element method used for solving a linearized Navier–Stokes problem in a time-dependent domain. In this study, the domain’s evolution is assumed to be known and independent of the solution to the problem at hand. The numerical method employed in the study combines a standard backward differentiation formula-type time-stepping procedure with a geometrically unfitted finite element discretization technique. Additionally, Nitsche’s method is utilized to enforce the boundary conditions. The paper presents a convergence estimate for several velocity–pressure elements that are inf-sup stable. The estimate demonstrates optimal order convergence in the energy norm for the velocity component and a scaled $$L^{2}(H^{1})$$-type norm for the pressure component. 

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2. Discontinuous Galerkin approximations to elliptic and parabolic problems with a Dirac line source

   https://doi.org/10.1051/m2an/2022095  

   Masri, Rami; Shen, Boqian; Riviere, Beatrice (March 2023, ESAIM: Mathematical Modelling and Numerical Analysis)

   The analyses of interior penalty discontinuous Galerkin methods of any order k for solving elliptic and parabolic problems with Dirac line sources are presented. For the steady state case, we prove convergence of the method by deriving a priori error estimates in the L 2 norm and in weighted energy norms. In addition, we prove almost optimal local error estimates in the energy norm for any approximation order. Further, almost optimal local error estimates in the L 2 norm are obtained for the case of piecewise linear approximations whereas suboptimal error bounds in the L 2 norm are shown for any polynomial degree. For the time-dependent case, convergence of semi-discrete and of backward Euler fully discrete scheme is established by proving error estimates in L 2 in time and in space. Numerical results for the elliptic problem are added to support the theoretical results. 

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3. Convergence and optimality of an adaptive modified weak Galerkin finite element method

   https://doi.org/10.1002/num.23027  

   Yingying, Xie; Cao, Shuhao; Chen, Long; Zhong, Liuqiang (April 2023, Numerical Methods for Partial Differential Equations)

   Abstract An adaptive modified weak Galerkin method (AmWG) for an elliptic problem is studied in this article, in addition to its convergence and optimality. The modified weak Galerkin bilinear form is simplified without the need of the skeletal variable, and the approximation space is chosen as the discontinuous polynomial space as in the discontinuous Galerkin method. Upon a reliable residual‐baseda posteriorierror estimator, an adaptive algorithm is proposed together with its convergence and quasi‐optimality proved for the lowest order case. The primary tool is to bridge the connection between the modified weak Galerkin method and the Crouzeix–Raviart nonconforming finite element. Unlike the traditional convergence analysis for methods with a discontinuous polynomial approximation space, the convergence of AmWG is penalty parameter free. Numerical results are presented to support the theoretical results. 

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4. Stability and error analysis of a semi-implicit scheme for incompressible flows with variable density and viscosity

   https://doi.org/10.1515/jnma-2024-0033  

   Vu, An; Cappanera, Loic (November 2024, Journal of Numerical Mathematics)

   Abstract We study the stability and convergence properties of a semi-implicit time stepping scheme for the incompressible Navier–Stokes equations with variable density and viscosity. The density is assumed to be approximated in a way that conserves the minimum-maximum principle. The scheme uses a fractional time-stepping method and the momentum, which is equal to the product of the density and velocity, as a primary unknown. The semi-implicit algorithm for the coupled momentum-pressure is shown to be conditionally stable and the velocity is shown to converge inL²norm with order one in time. Numerical illustrations confirm that the algorithm is stable and convergent under classic CFL condition even for sharp density profiles. 

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5. Can Second‐Order Numerical Accuracy Be Achieved for Moist Atmospheric Dynamics With Non‐Smoothness at Cloud Edge?

   https://doi.org/10.1029/2025MS005293  

   Marsico, David H; Du, Shukai; Stechmann, Samuel N (October 2025, Journal of Advances in Modeling Earth Systems)

   Non‐smoothness arises at cloud edge because, in moist thermodynamics, the thermodynamic properties of the atmosphere are different inside a cloud versus in clear air. In particular, inside a cloud, the vapor pressure of water is constrained by the saturation vapor pressure, which acts as a threshold. Due to this threshold, while the water vapor mixing ratio may vary continuously across cloud edge, its derivatives are not necessarily continuous at cloud edge. Similarly, non‐smoothness also arises for buoyancy and other variables. Consequently, this non‐smoothness in buoyancy and other variables can cause a degraded accuracy in computational simulations. Here we consider special treatment of numerical methods for the interface that arises from phase changes and cloud edges, in order to enhance the accuracy and potentially achieve second‐order accuracy. Numerical solutions are computed for the moist non‐precipitating Boussinesq equations as an idealized cloud‐resolving model with phase changes of water, that is, with cloud formation. Convergence tests, both spatial and temporal, are conducted to measure the numerical error as the grid spacing and time step are refined. While approximately second‐order accuracy is seen in root‐mean‐square (L2) error, the accuracy is degraded in the maximum (Linfinity) error. Discussion is also included on theoretical issues and potential implications for numerical simulations. 

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