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1. Optimally convergent mixed finite element methods for the stochastic Stokes equations

   https://doi.org/10.1093/imanum/drab006  

   Feng, Xiaobing; Prohl, Andreas; Vo, Liet (March 2021, IMA Journal of Numerical Analysis)

   null (Ed.)

   Abstract We propose some new mixed finite element methods for the time-dependent stochastic Stokes equations with multiplicative noise, which use the Helmholtz decomposition of the driving multiplicative noise. It is known (Langa, J. A., Real, J. & Simon, J. (2003) Existence and regularity of the pressure for the stochastic Navier--Stokes equations. Appl. Math. Optim., 48, 195--210) that the pressure solution has low regularity, which manifests in suboptimal convergence rates for well-known inf-sup stable mixed finite element methods in numerical simulations; see Feng X., & Qiu, H. (Analysis of fully discrete mixed finite element methods for time-dependent stochastic Stokes equations with multiplicative noise. arXiv:1905.03289v2 [math.NA]). We show that eliminating this gradient part from the noise in the numerical scheme leads to optimally convergent mixed finite element methods and that this conceptual idea may be used to retool numerical methods that are well known in the deterministic setting, including pressure stabilization methods, so that their optimal convergence properties can still be maintained in the stochastic setting. Computational experiments are also provided to validate the theoretical results and to illustrate the conceptual usefulness of the proposed numerical approach. 

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2. narrow-stencil framework for convergent numerical approximations of fully nonlinear second order PDEs

   https://doi.org/10.58997/ejde.conf.26.f1  

   Feng, Xiaobing; Lewis, Thomas; Ward, Kellie (June 2023, Electronic Journal of Differential Equations)

   This article develops a unified general framework for designing convergent finite difference and discontinuous Galerkin methods for approximating viscosity and regular solutions of fully nonlinear second order PDEs. Unlike the well-known monotone (finite difference) framework, the proposed new framework allows for the use of narrow stencils and unstructured grids which makes it possible to construct high order methods. The general framework is based on the concepts of consistency and g-monotonicity which are both defined in terms of various numerical derivative operators. Specific methods that satisfy the framework are constructed using numerical moments. Admissibility, stability, and convergence properties are proved,and numerical experiments are provided along with some computer implementation details. For more information see https://ejde.math.txstate.edu/conf-proc/26/f1/abstr.html 

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3. Reduced Order Models for the Quasi-Geostrophic Equations: A Brief Survey

   https://doi.org/10.3390/fluids6010016  

   Mou, Changhong; Wang, Zhu; Wells, David R.; Xie, Xuping; Iliescu, Traian (January 2021, Fluids)

   null (Ed.)

   Reduced order models (ROMs) are computational models whose dimension is significantly lower than those obtained through classical numerical discretizations (e.g., finite element, finite difference, finite volume, or spectral methods). Thus, ROMs have been used to accelerate numerical simulations of many query problems, e.g., uncertainty quantification, control, and shape optimization. Projection-based ROMs have been particularly successful in the numerical simulation of fluid flows. In this brief survey, we summarize some recent ROM developments for the quasi-geostrophic equations (QGE) (also known as the barotropic vorticity equations), which are a simplified model for geophysical flows in which rotation plays a central role, such as wind-driven ocean circulation in mid-latitude ocean basins. Since the QGE represent a practical compromise between efficient numerical simulations of ocean flows and accurate representations of large scale ocean dynamics, these equations have often been used in the testing of new numerical methods for ocean flows. ROMs have also been tested on the QGE for various settings in order to understand their potential in efficient numerical simulations of ocean flows. In this paper, we survey the ROMs developed for the QGE in order to understand their potential in efficient numerical simulations of more complex ocean flows: We explain how classical numerical methods for the QGE are used to generate the ROM basis functions, we outline the main steps in the construction of projection-based ROMs (with a particular focus on the under-resolved regime, when the closure problem needs to be addressed), we illustrate the ROMs in the numerical simulation of the QGE for various settings, and we present several potential future research avenues in the ROM exploration of the QGE and more complex models of geophysical flows. 

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4. A Universal Predictor‐Corrector Approach for Minimizing Artifacts Due To Mesh Refinement

   https://doi.org/10.1029/2023MS003688  

   Du, Shukai; Stechmann, Samuel N (November 2023, Journal of Advances in Modeling Earth Systems)

   Abstract With nested grids or related approaches, it is known that numerical artifacts can be generated at the interface of mesh refinement. Most of the existing methods of minimizing these artifacts are either problem‐dependent or numerical methods‐dependent. In this paper, we propose a universal predictor‐corrector approach to minimize these artifacts. By its construction, the approach can be applied to a wide class of models and numerical methods without modifying the existing methods but instead incorporating an additional step. The idea is to use an additional grid setup with a refinement interface at a different location, and then to correct the predicted state near the refinement interface by using information from the other grid setup. We give some analysis for our method in the setting of a one‐dimensional advection equation, showing that the key to the success of the method depends on an optimized way of choosing the weight functions, which determine the strength of the corrector at a certain location. Furthermore, the method is also tested in more general settings by numerical experiments, including shallow water equations, multi‐dimensional problems, and a variety of underlying numerical methods including finite difference/finite volume and spectral element. Numerical tests suggest the effectiveness of the method on reducing numerical artifacts due to mesh refinement. 

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5. Coupled and decoupled stabilized mixed finite element methods for nonstationary dual‐porosity‐Stokes fluid flow model

   https://doi.org/10.1002/nme.6158  

   Al Mahbub, Md. Abdullah; He, Xiaoming; Nasu, Nasrin Jahan; Qiu, Changxin; Zheng, Haibiao (July 2019, International Journal for Numerical Methods in Engineering)

   Summary In this paper, we propose and analyze two stabilized mixed finite element methods for the dual‐porosity‐Stokes model, which couples the free flow region and microfracture‐matrix system through four interface conditions on an interface. The first stabilized mixed finite element method is a coupled method in the traditional format. Based on the idea of partitioned time stepping, the four interface conditions, and the mass exchange terms in the dual‐porosity model, the second stabilized mixed finite element method is decoupled in two levels and allows a noniterative splitting of the coupled problem into three subproblems. Due to their superior conservation properties and convenience of the computation of flux, mixed finite element methods have been widely developed for different types of subsurface flow problems in porous media. For the mixed finite element methods developed in this article, no Lagrange multiplier is used, but an interface stabilization term with a penalty parameter is added in the temporal discretization. This stabilization term ensures the numerical stability of both the coupled and decoupled schemes. The stability and the convergence analysis are carried out for both the coupled and decoupled schemes. Three numerical experiments are provided to demonstrate the accuracy, efficiency, and applicability of the proposed methods. 

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