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1. A Modified Brinkman Penalization Fictitious Domain Method for the Unsteady Navier‐Stokes Equations

   https://doi.org/10.1002/num.70089  

   Baitulenov, Zhanybek; Olshanskii, Maxim; Temirbekov, Almas; Temirbekov, Nurlan; Kasenov, Syrym (May 2026, Numerical Methods for Partial Differential Equations)

   ABSTRACT This paper investigates a modification of the fictitious domain method with continuation in the lower‐order coefficients for the unsteady Navier‐Stokes equations governing the motion of an incompressible homogeneous fluid in a bounded 2D or 3D domain. The modification enables a solution‐dependent choice of the critical parameter. Global‐in‐time existence and convergence of a weak solution to the auxiliary problem are proved, and local‐in‐time existence and convergence of a unique strong solution are established. For the strong solution, a new higher‐order convergence rate estimate in the penalization parameter is obtained. The introduced framework allows us to apply a pointwise divergence‐free finite element method as a discretization technique, leading to a strongly mass conservative discrete fictitious domain method. A numerical example illustrates the performance of the method. 

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2. Variational data assimilation with finite-element discretization for second-order parabolic interface equation

   https://doi.org/10.1093/imanum/drae010  

   Li, Xuejian; He, Xiaoming; Gong, Wei; Douglas, Craig C (May 2024, IMA Journal of Numerical Analysis)

   Abstract In this paper, we propose and analyze a finite-element method of variational data assimilation for a second-order parabolic interface equation on a two-dimensional bounded domain. The Tikhonov regularization plays a key role in translating the data assimilation problem into an optimization problem. Then the existence, uniqueness and stability are analyzed for the solution of the optimization problem. We utilize the finite-element method for spatial discretization and backward Euler method for the temporal discretization. Then based on the Lagrange multiplier idea, we derive the optimality systems for both the continuous and the discrete data assimilation problems for the second-order parabolic interface equation. The convergence and the optimal error estimate are proved with the recovery of Galerkin orthogonality. Moreover, three iterative methods, which decouple the optimality system and significantly save computational cost, are developed to solve the discrete time evolution optimality system. Finally, numerical results are provided to validate the proposed method. 

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3. A Lagrange multiplier formulation for the fully dynamic Navier–Stokes–Biot system

   https://doi.org/10.1093/imanum/draf074  

   Wang, Xing; Yotov, Ivan (September 2025, IMA Journal of Numerical Analysis)

   Abstract We study a mathematical model of fluid–poroelastic structure interaction and its numerical solution. The free fluid region is governed by the unsteady incompressible Navier–Stokes equations, while the poroelastic region is modelled by the Biot system of poroelasticity. The two systems are coupled along an interface through continuity of normal velocity and stress and the Beavers–Joseph–Saffman slip with friction condition. The variables in the weak formulation are velocity and pressure for Navier–Stokes, displacement for elasticity and velocity and pressure for Darcy flow. A Lagrange multiplier of stress/pressure type is employed to impose weakly the continuity of flux. Existence, uniqueness and stability of a weak solution is established under a small data assumption. A fully discrete numerical method is then developed, based on backward Euler time discretization and finite element spatial approximation. We establish solvability, stability and error estimates for the fully discrete scheme. Numerical experiments are presented to verify the theoretical results and illustrate the performance of the method for an arterial flow application. 

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4. A Coupled Method Combining Crouzeix-Raviart Nonconforming and Node Conforming Finite Element Spaces for Boit Consolidation Model

   https://doi.org/10.4208/jcm.2212-m2021-0231  

   Zeng, Yuping; Cai, Mingchao; Zhong, Liuqiang (April 2024, Journal of Computational Mathematics)

   A mixed finite element method is presented for the Biot consolidation problem in poroelasticity. More precisely, the displacement is approximated by using the Crouzeix-Raviart nonconforming finite elements, while the fluid pressure is approximated by using the node conforming finite elements. The well-posedness of the fully discrete scheme is established, and a corresponding priori error estimate with optimal order in the energy norm is also derived. Numerical experiments are provided to validate the theoretical results. 

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5. An efficient flux‐variable approximation scheme for Darcy's flow

   https://doi.org/10.1002/num.23120  

   Adhikari, Rajan B; Kim, Imbumn; Lee, Young Ju; Sheen, Dongwoo (June 2024, Numerical Methods for Partial Differential Equations)

   Abstract We present an efficient numerical method to approximate the flux variable for the Darcy flow model. An important feature of our new method is that the approximate solution for the flux variable is obtained without approximating the pressure at all. To accomplish this, we introduce a user‐defined parameter delta, which is typically chosen to be small so that it minimizes the negative effect resulting from the absence of the pressure, such as inaccuracy in both the flux approximation and the mass conservation. The resulting algebraic system is of significantly smaller degrees of freedom, compared to the one from the mixed finite element methods or least‐squares methods. We also interpret the proposed method as a single step iterate of the augmented Lagrangian Uzawa applied to solve the mixed finite element in a special setting. Lastly, the pressure recovery from the flux variable is discussed and an optimal‐order error estimate for the method is obtained. Several examples are provided to verify the proposed theory and algorithm, some of which are from more realistic models such as SPE10. 

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