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This paper demonstrates the mixed formulation of the Brinkman problem using linear equal-order finite element methods in porous media modelling. We introduce Galerkin least-squares (GLS) and least-squares (LS) finite element methods to address the incompatibility of finite element spaces, treating velocity, pressure, and vorticity as independent variables. Theoretical analysis examines coercivity and continuity, providing error estimates. Demonstrating resilience in theoretical findings, these methods achieve optimal convergence rates in the $ L^2 $ L 2 norm by incorporating stabilization terms with low-order basis functions. Numerical experiments validate theoretical predictions, showing the effectiveness of the GLS method and addressing finite element space incompatibility. Additionally, the GLS method exhibits promising capabilities in handling the Brinkman equation at low permeability compared to the LS method. The study reveals an increase in the average pressure difference in the Brinkman problem compared to the Stokes equations as the inlet velocity rises, providing insights into the behaviour of Brinkman equations. [ABSTRACT FROM AUTHOR]

In this paper, we develop a least-squares (LS) finite element method for the Stokes equations. The LS method uses the-norm of the residuals of the continuity equation multiplied by appropriately adjusted weights. To adjust the weight, we employ an adaptive weight iteration approach based on a residual type a posteriori error estimator for the LS functional. This LS method is applied to flows through a planar channel, lid-driven cavity flows, and flows past a transverse slot. The results demonstrate that mass conservation of the LS method in fluid simulations can be significantly improved by appropriately adjusting weights, and that this can be accomplished using low-order basis functions, without substantial complications in coding. We provide a posteriori error estimates for the linearized velocity–vorticity–pressure first-order system and show numerical results supporting the estimate. Numerical results reveal that the mass conservation constant is problem dependent. [ABSTRACT FROM AUTHOR]

A framework for solving the non-isothermal unsteady Reynolds-averaged Navier-Stokes equations with emphasis on applications to thermal building simulation is prescribed in this paper. Different domain decomposition techniques are used (i) for the treatment of boundary layers, (ii) for the efficient solution of the arising linear subproblems, and (iii) for coupling the indoor air flow field with the ambient. The approach is then applied to exemplary indoor air flow configurations. [ABSTRACT FROM AUTHOR]

In this computational study, we deal with the stabilized finite element solutions of convection-dominated models having nonlinear reaction mechanisms. The existence of advection terms in these models causes the numerical solutions obtained by standard discretization methods to exhibit nonphysical oscillations. The Galerkin finite element method is stabilized using the Streamline-Upwind/Petrov–Galerkin formulation to avoid such spurious oscillations. The stabilized formulation is also complemented with the YZβ shock-capturing technique to resolve steep gradients and discontinuities accurately. The nonlinear equation systems arising from the spatial discretizations are solved with the Newton–Raphson method supplemented with the ILU-preconditioned GMRES search technique. The proposed methods are tested on a comprehensive set of nonlinear reactive models. Numerical experiments show that the presented methods and techniques eliminate spurious oscillations significantly and resolve strong gradients accurately. All computations are performed in the FEniCS environment. [ABSTRACT FROM AUTHOR]

In this study, a least-squares (LS) finite element method with an adaptive mesh approach is investigated for Giesekus viscoelastic flow problems. We consider the weighted LS method on uniform and adaptive meshes for the Newton linearized viscoelastic problem, where adaptive grids are automatically generated by the least-squares solutions. We use a residual-type a-posteriori error estimator to adjust weights in the LS functional and compare the convergence behaviour of adaptive meshes generated using different grading functions. Numerical results demonstrate that the adaptive LS method shows at least the first-order convergence rate when equal-order linear interpolation functions are used for all variables, which agrees with the theoretical estimate. In addition, adaptive grids generated using the velocity outperform those based on the a-posteriori error estimator, yielding better numerical results. [ABSTRACT FROM AUTHOR]

We implemented weighted least-squares finite element methods for the linearized Navier-Stokes equations based on the velocity–pressure–stress and the velocity–vorticity–pressure formulations. The least-squares functionals involve theL2-norms of the residuals of each equation multiplied by the appropriate weighting functions. The weights included a mass conservation constant, a mesh-dependent weight, a nonlinear weighting function, and Reynolds numbers. A feature of this approach is that the linearized system creates a symmetric and positive-definite linear algebra problem at each Newton iteration. We can prove that least-squares approximations converge with the linearized version solutions of the Navier–Stokes equations at the optimal convergence rate. Model problems considered in this study were the flow past a planar channel and 4-to-1 contraction problems. We presented approximate solutions of the Navier–Stokes problems by solving a sequence of the linearized Navier–Stokes problems arising from Newton iterations, revealing the convergence rates of the proposed schemes, and investigated Reynolds number effects. [ABSTRACT FROM AUTHOR]

The major emphasis of this work is the development of a stabilized finite element method for solving incompressible Navier-Stokes equations with stochastic input data. The polynomial chaos expansion is used to represent stochastic processes in the variational problem, resulting in a set of deterministic variational problems to be solved for each Wiener polynomial chaos. To obtain the chaos coefficients in the corresponding deterministic incompressible Navier-Stokes equations, we combine the modified method of characteristics with the finite element discretization. The obtained Stokes problem is solved using a robust conjugate-gradient algorithm. This algorithm avoids projection procedures and any special correction for the pressure. These numerical techniques associate the geometrical flexibility of the finite element method with the ability offered by the modified method of characteristics to solve convection-dominated problems using time steps larger than its Eulerian counterpart. Numerical results are shown for (he benchmark problems of driven cavity flow and backward-facing step flow. We also present numerical results for a problem of stochastic natural convection. It is found that the proposed stabilized finite element method offers a robust and accurate approach for solving the stochastic incompressible Navier-Stokes equations, even when high Reynolds and Rayleigh numbers are used in the simulations. [ABSTRACT FROM AUTHOR]