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In this paper, a two-level Newton iterative method is proposed for the stationary micropolar fluid equations. Firstly, the original equations are solved on a coarse grid based on Newton-type linearization. Then, the simplified linearized equations are solved on a fine grid. The stability and error estimates of the method are given in the theoretical part. The results of the theoretical analysis show that when the coarse mesh size H and fine mesh size h satisfy the relation h = O (H 2) , the two-level Newton iterative method can achieve an optimal convergence rate. Finally, the effectiveness and applicability of the method are verified by some numerical experiments. [ABSTRACT FROM AUTHOR]

In this paper, we propose and analyze two stabilized finite element schemes for the Stokes–Darcy-transport model. The first stabilized method is a monolithic scheme with the backward-Euler time discretization, and fully coupled by one Stokes subproblem, one Darcy subproblem, and two transport subproblems. The second stabilized method is a non-iterative partitioned scheme which splits the fully coupled transport problem into two decoupled subproblems. The stability of the proposed schemes can be ensured by some strongly consistent interface terms. The numerical experiments are performed to illustrate the theoretical analysis and demonstrate the reliability and applicability of the proposed schemes. [ABSTRACT FROM AUTHOR]

This paper is devoted to the study of Bingham flow with variable density. We propose a local bi-viscosity regularization of the stress tensor based on a Huber smoothing step. Next, our computational approach is based on a second-order, divergence-conforming discretization of the Huber regularized Bingham constitutive equations, coupled with a discontinuous Galerkin scheme for the mass density. We take advantage of the properties of divergence-conforming and discontinuous Galerkin formulations to effectively incorporate upwind discretizations, thereby ensuring the stability of the formulation. The stability of the continuous problem and the fully discrete scheme are analyzed. Further, a semismooth Newton method is proposed for solving the obtained fully discretized system of equations at each time step. Finally, several numerical examples that illustrate the main features of the problem and the properties of the numerical scheme are presented. [ABSTRACT FROM AUTHOR]

In this paper, we propose and analyze a new structure-preserving finite element method for the stationary magnetohydrodynamic equations with magnetic-current formulation on Lipschitz domains. Using a mixed finite element approach, we discretize the hydrodynamic unknowns by inf-sup stable velocity-pressure finite element pairs, and the current density, the induced electric field and the magnetic field by using the edge-edge-face elements from a discrete de-Rham complex pair. To deal with the divergence-free condition of the magnetic field, we introduce an augmented term to the discrete scheme rather than a Lagrange multiplier in the existing schemes. Thanks to discrete differential forms and finite element exterior calculus, the proposed scheme preserves the divergence-free property exactly for the magnetic induction on the discrete level. The well-posedness of the discrete problem is further proved under the small data condition. Under weak regularity assumptions, we rigorously establish the error estimates of the finite element schemes. Numerical results are provided to illustrate the theoretical results and demonstrate the efficiency of the proposed method. [ABSTRACT FROM AUTHOR]

In this paper, pressure projection stabilized low-order mixed finite element methods are studied to solve a Navier-Stokes hemivariational inequality for a boundary value problem of the Navier-Stokes equations involving a non-smooth non-monotone boundary condition. A new abstract mixed hemivariational inequality is introduced for the purpose of analyzing stabilized mixed finite element methods to solve the Navier-Stokes hemivariational inequality using velocity-pressure pairs without the discrete inf-sup condition. The well-posedness of the abstract problem is established through considerations of a related saddle-point formulation and fixed-point arguments. Then the results on the abstract problem are applied to the study of the Navier-Stokes hemivariational inequality and its stabilized mixed finite element approximations. Optimal order error estimates are derived for finite element solutions of the pressure projection stabilized lowest-order conforming pair and lowest equal order pair under appropriate solution regularity assumptions. Numerical results are reported on the performance of the pressure projection stabilized mixed finite element methods for solving the Navier-Stokes hemivariational inequality. [ABSTRACT FROM AUTHOR]

In this paper we consider the Navier–Stokes–Brinkman equations, which constitute one of the most common nonlinear models utilized to simulate viscous fluids through porous media, and propose and analyze a Banach spaces-based approach yielding new mixed finite element methods for its numerical solution. In addition to the velocity and pressure, the strain rate tensor, the vorticity, and the stress tensor are introduced as auxiliary unknowns, and then the incompressibility condition is used to eliminate the pressure, which is computed afterwards by a postprocessing formula depending on the stress and the velocity. The resulting continuous formulation becomes a nonlinear perturbation of, in turn, a perturbed saddle point linear system, which is then rewritten as an equivalent fixed-point equation whose operator involved maps the velocity space into itself. The well-posedness of it is then analyzed by applying the classical Banach fixed point theorem, along with a smallness assumption on the data, the Babuška–Brezzi theory in Banach spaces, and a slight variant of a recently obtained solvability result for perturbed saddle point formulations in Banach spaces as well. The resulting Galerkin scheme is momentum-conservative. Its unique solvability is analyzed, under suitable hypotheses on the finite element subspaces, using a similar fixed-point strategy as in the continuous problem. A priori error estimates are rigorously derived, including also that for the pressure. We show that PEERS and AFW elements for the stress, the velocity, and the rotation, together with piecewise polynomials of a proper degree for the strain rate tensor, yield stable discrete schemes. Then, the approximation properties of these subspaces and the Céa estimate imply the respective rates of convergence. Finally, we include two and three dimensional numerical experiments that serve to corroborate the theoretical findings, and these tests illustrate the performance of the proposed mixed finite element methods. [ABSTRACT FROM AUTHOR]

This paper aims to improve guaranteed error control for the Stokes problem with a focus on pressure-robustness, i.e., for discretisations that compute a discrete velocity that is independent of the exact pressure. A Prager–Synge type result relates the velocity errors of divergence-free primal and perfectly equilibrated dual mixed methods for the velocity stress. The first main result of the paper is a framework with relaxed constraints on the primal and dual method. This enables to use a recently developed mass conserving mixed stress discretisation for the design of equilibrated fluxes and to obtain pressure-independent guaranteed upper bounds for any pressure-robust (not necessarily divergence-free) primal discretisation. The second main result is a provably efficient local design of the equilibrated fluxes with comparably low numerical costs. Numerical examples verify the theoretical findings and show that efficiency indices of our novel guaranteed upper bounds are close to one. [ABSTRACT FROM AUTHOR]

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]

Based on local Gauss integral technique and backtracking technique, this paper presents and studies three kinds of two-grid stabilized finite element algorithms for the stationary Navier-Stokes equations. The proposed methods consist of deducing a coarse solution on the nonlinear system, updating the solution on a fine mesh via three different methods, and solving a linear correction problem on the coarse mesh to obtain the final solution. The error estimates are derived for the solution approximated by the proposed algorithms. A series of numerical experiments are illustrated to test the applicability and efficiency of our proposed methods, and support the theoretical analysis results. [ABSTRACT FROM AUTHOR]

In this paper, the rotational pressure-correction schemes for the closed-loop geothermal system are developed and analyzed. The primary benefit of this projection method is to replace the incompressible condition. The system is considered consisting of two distinct regions, with the free flow region governed by the Navier–Stokes equations and the porous media region governed by Darcy’s law. At the same time, the heat equations are coupled with the flow equations to describe the heat transfer in both regions. In the closed-loop geothermal system, the rotational pressure-correction schemes are used for the Navier–Stokes equations in the free flow region, and the direct decoupled scheme is used for the other equations. Besides, the stability of the proposed methods is proved. Finally, the high efficiency and applicability of the decoupled scheme are verified by 2D/3D numerical experiments. [ABSTRACT FROM AUTHOR]

In this paper we analyze a virtual element method for the two dimensional elasticity problem allowing small edges. With this approach, the classic assumptions on the geometrical features of the polygonal meshes can be relaxed. In particular, we consider only star-shaped polygons for the meshes. Suitable error estimates are presented, where a rigorous analysis on the influence of the Lamé constants in each estimate is presented. We report numerical tests to assess the performance of the method. [ABSTRACT FROM AUTHOR]

The paper presents the dynamical core of the new sea ice model SIMUG (Sea Ice Model on Unstructured Grid) on the A- and CD-types of unstructured triangular grids in the local-element basis on sphere. Three standardized box tests to reproduce the Linear Kinematic Features (LKFs), and the short-term forecast in the real Arctic Ocean geometry with the realistic atmosphere and ocean forcing demonstrate the model quality compared to other sea ice models like CICE, FESOM, MITgcm, and ICON-O. The distinctive features of the model presented are a wide choice of transport schemes, and the new numerical implementation with the serial and parallel C++ coding and INMOST, Ani2D, and Ani3D packages to deal with unstructured grids. Code profiling and scalability assessment are carried out. In general, the A-version of the ice drift model works faster, but has fewer degrees of freedom on the same grid. Due to the increase in the degrees of freedom, the model on the CD grid gives ultra-resolution of LKFs, but requires more strict conditions for stability. [ABSTRACT FROM AUTHOR]

This paper constructs and analyzes a boundary correction finite element method for the Stokes problem based on the Scott–Vogelius pair on Clough–Tocher splits. The velocity space consists of continuous piecewise polynomials of degree k, and the pressure space consists of piecewise polynomials of degree (k – 1) without continuity constraints. A Lagrange multiplier space that consists of continuous piecewise polynomials with respect to the boundary partition is introduced to enforce boundary conditions and to mitigate the lack of pressure-robustness. We prove several inf-sup conditions, leading to the well-posedness of the method. In addition, we show that the method converges with optimal order and the velocity approximation is divergence-free. [ABSTRACT FROM AUTHOR]

In this paper we consider a steady phase change problem for non-isothermal incompressible viscous flow in porous media with an enthalpy-porosity-viscosity coupling mechanism, and introduce and analyze a Banach spaces-based variational formulation yielding a new mixed-primal finite element method for its numerical solution. The momentum and mass conservation equations are formulated in terms of velocity and the tensors of strain rate, vorticity, and stress; and the incompressibility constraint is used to eliminate the pressure, which is computed afterwards by a postprocessing formula depending on the stress and the velocity. The resulting continuous formulation for the flow becomes a nonlinear perturbation of a perturbed saddle point linear system. The energy conservation equation is written as a nonlinear primal formulation that incorporates the additional unknown of boundary heat flux. The whole mixed-primal formulation is regarded as a fixed-point operator equation, so that its well-posedness hinges on Banach’s theorem, along with smallness assumptions on the data. In turn, the solvability analysis of the uncoupled problem in the fluid employs the Babuška–Brezzi theory, a recently obtained result for perturbed saddle-point problems, and the Banach–Nečas–Babuška Theorem, all them in Banach spaces, whereas the one for the uncoupled energy equation applies a nonlinear version of the Babuška–Brezzi theory in Hilbert spaces. An analogue fixed-point strategy is employed for the analysis of the associated Galerkin scheme, using in this case Brouwer’s theorem and assuming suitable conditions on the respective discrete subspaces. The error analysis is conducted under appropriate assumptions, and selecting specific finite element families that fit the theory. We finally report on the verification of theoretical convergence rates with the help of numerical examples. [ABSTRACT FROM AUTHOR]

The construction of weak solutions to compressible Navier–Stokes equations via a numerical method (including a rigorous proof of the convergence) is in a short supply, and so far, available only for one sole numerical scheme suggested in Karper (Numer Math, 125(3):441–510, 2013) for the no slip boundary conditions and the isentropic pressure with adiabatic coefficient γ > 3 . Here we consider the same problem for the general non zero inflow–outflow boundary conditions, which is definitely more appropriate setting from the point of view of applications, but which is essentially more involved as far as the existence of weak solutions is concerned. There is a few recent proofs of existence of weak solutions in this setting, but none of them is performed via a numerical method. The goal of this paper is to fill this gap. The existence of weak solutions on the continuous level requires several tools of functional and harmonic analysis and differential geometry whose numerical counterparts are not known. Our main strategy therefore consists in rewriting of the numerical scheme in its variational form modulo remainders and to apply and/or to adapt to the new variational formulation the tools developed in the theoretical analysis. In addition to the result, which is new, the synergy between numerical and theoretical analysis is the main originality of the present paper. [ABSTRACT FROM AUTHOR]

In this paper we analyze a fully discrete scheme for a general Cahn–Hilliard equation coupled with a nonsteady Magneto-hydrodynamics flow, which describes two immiscible, incompressible and electrically conducting fluids with different mobilities, fluid viscosities and magnetic diffusivities. A typical fully discrete scheme, which is comprised of conforming finite element method and the Euler semi-implicit discretization based on a convex splitting of the energy of the equation is considered in detail. We prove that our scheme is unconditionally energy stable and obtain some optimal error estimates for the concentration field, the chemical potential, the velocity field, the magnetic field and the pressure. The results of numerical tests are presented to validate the rates of convergence. [ABSTRACT FROM AUTHOR]

This paper introduces two methods for the fully discrete time-dependent Bingham problem in a three-dimensional domain and for the flow in a pipe also named after Mosolov. The first time discretisation is a generalised midpoint rule and the second time discretisation is a discontinuous Galerkin scheme. The space discretisation in both cases employs the non-conforming first-order finite elements of Crouzeix and Raviart. The a priori error analyses for both schemes yield certain convergence rates in time and optimal convergence rates in space. It guarantees convergence of the fully-discrete scheme with a discontinuous Galerkin time-discretisation for consistent initial conditions and a source term f ∈ H 1 (0 , T ; L 2 (Ω)) . [ABSTRACT FROM AUTHOR]

The shallow water equations (SWE) are a time-dependent system of non-linear partial differential equations utilized for fluid motion where the horizontal length scales are much greater than the fluid depth. The numerical solution of the SWE usually requires complex schemes and methods to deal with the instabilities proper to the system. This paper is concerned with a simple method to solve the SWE, utilizing continuous linear finite elements, no-slip closed boundary, and nested open boundary conditions, and a stabilization technique known as numerical smoothing. First, we describe the governing equations, the finite element model, and the chosen discretization. Then we solve a 2D test case of an elliptical paraboloid and analyze its convergence comparing with provided analytical solutions. Finally, we apply the model to the northern region of the Gulf of San Jorge (Argentina), optimize parameters, and validate it with collected data from an acoustic Doppler current profiler (ADCP) of two spatial points during one tidal M 2 period. [ABSTRACT FROM AUTHOR]

In this paper, we propose a novel difference finite element (DFE) method based on the P1-element for the 3D heat equation on a 3D bounded domain. One of the novel ideas of this paper is to use the second-order backward difference formula (BDF) combining DFE method to overcome the computational complexity of conventional finite element (FE) method for the high-dimensional parabolic problem. First, we design a fully discrete difference FE solution u h n by the second-order backward difference formula in the temporal t-direction, the center difference scheme in the spatial z-direction, and the P1-element on a almost-uniform mesh Jh in the spatial (x, y)-direction. Next, the H1-stability of u h n and the second-order H1-convergence of the interpolation post-processing function on u h n with respect to u(tn) are provided. Finally, numerical tests are presented to show the second-order H1-convergence results of the proposed DFE method for the heat equation in a 3D spatial domain. [ABSTRACT FROM AUTHOR]

In this paper, we present a numerical analysis of the hydrostatic Stokes equations, which are linearization of the primitive equations describing the geophysical flows of the ocean and the atmosphere. The hydrostatic Stokes equations can be formulated as an abstract non-stationary saddle-point problem, which also includes the non-stationary Stokes equations. We first consider the finite element approximation for the abstract equations with a pair of spaces under the discrete inf-sup condition. The aim of this paper is to establish error estimates for the approximated solutions in various norms, in the framework of analytic semigroup theory. Our main contribution is an error estimate for the pressure with a natural singularity term t-1, which is induced by the analyticity of the semigroup. We also present applications of the error estimates for the finite element approximations of the non-stationary Stokes and the hydrostatic Stokes equations. [ABSTRACT FROM AUTHOR]

In this paper we analyze a posteriori error estimates for a mixed formulation of the linear elasticity eigenvalue problem. A posteriori estimators for the nearly and perfectly compressible elasticity spectral problems are proposed. With a post-process argument, we are able to prove reliability and efficiency for the proposed estimators. The numerical method is based in Raviart-Thomas elements to approximate the pseudostress and piecewise polynomials for the displacement. We illustrate our results with numerical tests in two and three dimensions. [ABSTRACT FROM AUTHOR]

In this paper, we propose and analyze a mixed finite element scheme for stationary inductionless magnetohydrodynamic equations on a general Lipschitz domain. We adopt divergence-conforming elements for the velocity and the current density, discontinuous elements for the pressure and the electric potential, thus the approximations for velocity and current density are exactly divergence-free. The H 1 -continuity of the velocity is enforced by discontinuous Galerkin approach. With this discretization, we prove the well-posedness of the discrete scheme, and derive optimal error estimates of the discrete solutions. In particular, we show that the error estimates for the velocity and the current density are independent of the pressure and the electric potential, and the error estimates for the pressure and the electric potential are also unrelated to each other. Based on this, we propose two coupled iterative methods: Stokes and Oseen iterations. Rigorous analysis of convergence and stability is provided. Finally, some numerical examples are performed to verify the theoretical results and show the effectiveness of the presented methods. [ABSTRACT FROM AUTHOR]

An a priori analysis for a generalized local projection stabilized finite element solution of the Darcy equations is presented in this paper. A first-order nonconforming ℙ 1 n c finite element space is used to approximate the velocity, whereas the pressure is approximated using two different finite elements, namely piecewise constant ℙ 0 and piecewise linear nonconforming ℙ 1 n c elements. The considered finite element pairs, ℙ 1 n c / ℙ 0 and ℙ 1 n c / ℙ 1 n c , are inconsistent and incompatibility, respectively, for the Darcy problem. The stabilized discrete bilinear form satisfies an inf-sup condition with a generalized local projection norm. Moreover, a priori error estimates are established for both finite element pairs. Finally, the validation of the proposed stabilization scheme is demonstrated with appropriate numerical examples. [ABSTRACT FROM AUTHOR]

Adaptive time stepping is an important tool for controlling the simulation error and enhancing its efficiency in computational fluid dynamics. In this paper, we introduce two adaptive time-stepping algorithms based on the scalar auxiliary variable scheme for the Navier-Stokes equation. One way is to take equidistribution of energy decay to control the variance of the time steps, and in the other way, the relative error is involved to produce time steps. The numerical experiments show that the computational time is significantly saved by these methods. [ABSTRACT FROM AUTHOR]

In this paper, we present two stable virtual element methods for the Brinkman equations robust in the Stokes and Darcy limits. We use a pair of stable virtual elements for the velocity and pressure. Firstly, we define an H (div) -conforming velocity reconstruction operator for the velocity test function. By employing it in the discretizations of the zero-order term and the right-hand-side source term, we propose the first virtual element method on convex polygonal meshes. Secondly, we construct the enhanced virtual element space in place of the original virtual element space such that one can exactly compute the L 2 -projection of virtual function onto linear polynomial space. We make full use of this projection to introduce the second virtual element method on general polygonal meshes. The well-posedness and uniform energy-error estimates of each method are strictly established. Finally, numerical experiments are provided to validate the theoretical results and illustrate the good performance of our methods. [ABSTRACT FROM AUTHOR]

Complex processes in perforated domains occur in many real-world applications. These problems are typically characterized by physical processes in domains with multiple scales. Moreover, these problems are intrinsically multiscale and their discretizations can yield very large linear or nonlinear systems. In this paper, we investigate multiscale approaches that attempt to solve such problems on a coarse grid by constructing multiscale basis functions in each coarse grid, where the coarse grid can contain many perforations. In particular, we are interested in cases when there is no scale separation and the perforations can have different sizes. In this regard, we mention some earlier pioneering works, where the authors develop multiscale finite element methods. In our paper, we follow Generalized Multiscale Finite Element Method (GMsFEM) and develop a multiscale procedure where we identify multiscale basis functions in each coarse block using snapshot space and local spectral problems. We show that with a few basis functions in each coarse block, one can approximate the solution, where each coarse block can contain many small inclusions. We apply our general concept to (1) Laplace equation in perforated domains; (2) elasticity equation in perforated domains; and (3) Stokes equations in perforated domains. Numerical results are presented for these problems using two types of heterogeneous perforated domains. The analysis of the proposed methods will be presented elsewhere. [ABSTRACT FROM AUTHOR]

In this paper, we study the a posteriori error estimate corresponding to the Brinkman–Darcy–Forchheimer problem. We introduce the variational formulation discretized by using the finite element method. Then, we establish an a posteriori error estimation with two types of error indicators related to the discretization and to the linearization. Finally, numerical investigations are shown and discussed. [ABSTRACT FROM AUTHOR]

This paper considers charge-conservative finite element approximation and three coupled iterations of stationary inductionless magnetohydrodynamics equations in Lipschitz domain. Using a mixed finite element method, we discretize the hydrodynamic unknowns by stable velocity-pressure finite element pairs, discretize the current density and electric potential by H (div , Ω) × L 2 (Ω) -comforming finite element pairs. The well-posedness of this formula and the optimal error estimate are provided. In particular, we show that the error estimates for the velocity, the current density and the pressure are independent of the electric potential. With this, we propose three coupled iterative methods: Stokes, Newton and Oseen iterations. Rigorous analysis of convergence and stability for different iterative schemes are provided, in which we improve the stability conditions for both Stokes and Newton iterative method. Numerical results verify the theoretical analysis and show the applicability and effectiveness of the proposed methods. [ABSTRACT FROM AUTHOR]

This paper considers Stokes and Newton iterations to solve stationary Navier- Stokes equations based on the finite element discretization. We obtain new sufficient conditions of stability and convergence for the two iterations. Specifically, when 0 < σ = $\tfrac{{N||f||_{ - 1} }} {{\nu ^2 }}$≤ $$\tfrac{1}{{\sqrt 2 + 1}}$$, the Stokes iteration is stable and convergent, where N is defined in the paper. When 0 < σ ≤ 5/11, the Newton iteration is stable and convergent. This work gives a more accurate admissible range of data for stability and convergence of the two schemes, which improves the previous results. A numerical test is given to verify the theory. [ABSTRACT FROM AUTHOR]

The idea of using velocity dilation for shock capturing is revisited in this paper, combined with the discontinuous Galerkin method. The value of artificial viscosity is determined using direct dilation instead of its higher order derivatives to reduce cost and degree of difficulty in computing derivatives. Alternative methods for estimating the element size of large aspect ratio and smooth artificial viscosity are proposed to further improve robustness and accuracy of the model. Several benchmark tests are conducted, ranging from subsonic to hypersonic flows involving strong shocks. Instead of adjusting empirical parameters to achieve optimum results for each case, all tests use a constant parameter for the model with reasonable success, indicating excellent robustness of the method. The model is only limited to third-order accuracy for smooth flows. This limitation may be relaxed by using a switch or a wall function. Overall, the model is a good candidate for compressible flows with potentials of further improvement. [ABSTRACT FROM AUTHOR]

Reactive transport modelling (RTM) is a powerful tool for understanding subsurface systems where fluid flow and chemical reactions occur simultaneously. RTM has been widely used to understand the formation of dolomite by replacement of calcite, which can be an important control on carbonate reservoir quality. Dolomitisation is a reactive transport process governed by slow dolomite precipitation and cannot be correctly simulated without a kinetic rate model. The new CSMP++GEM coupled RTM code uses the GEMS3K kernel for solving geochemical equilibria by the Gibbs energy minimisation method with the CSMP++ framework that implements a hybrid finite element-finite volume method to solve partial differential equations. The unique feature of the new coupling is the mineral reaction kinetics, implemented via additional metastability constraints. CSMP++GEM is able to simulate single-phase flow and solute transport in porous media together with chemical reactions at different pressure, temperature, and water salinity conditions. This RTM assures mass conservation which is crucial when simulating transport of solutes with low concentrations over geological time. A full feedback of mineral dissolution/precipitation on the fluid flow is provided via corresponding porosity/permeability evolution and two source terms in the pressure equation. First, the mass source term accounts for the mass of solutes released during mineral dissolution or taken from the solution by mineral precipitation. The second source term attributes to the fact that the solution density is affected by mineral dissolution/precipitation, too. This effect is included through the equivalent water salinity, which is calculated from the total amount of dissolved solutes and is used to update the properties of saline water from the H $$_2$$ O-NaCl equation of state. This paper puts emphasis on the thorough mathematical derivation of the governing equations and a detailed description of the numerical solution procedure. Two sets of benchmarking results are presented. The first benchmark is a well-known 1D model of dolomitisation by MgCl $$_2$$ solution with thermodynamic reactions. In the second benchmark, CSMP++GEM is compared with TOUGHREACT on a 1D model of dolomitisation by sea water taking into account mineral reaction kinetics. The results presented in this paper demonstrate the ability of the CSMP++GEM code to correctly reproduce dolomitisation effects. [ABSTRACT FROM AUTHOR]

The paper discusses a stabilization of a finite element method for the equations of fluid motion in a time-dependent domain. After experimental convergence analysis, the method is applied to simulate a blood flow in the right ventricle of a post-surgery patient with the transposition of the great arteries disorder. The flow domain is reconstructed from a sequence of 4D CT images. The corresponding segmentation and triangulation algorithms are also addressed in brief. [ABSTRACT FROM AUTHOR]

The Peterlin viscoelastic model describes the motion of certain incompressible polymeric fluids. It employs a nonlinear dumbbell model with a nonlinear spring force law making it more nonlinear than other viscoelastic models. In this paper, we propose and study a fully implicit stabilized Crank-Nicolson time stepping scheme for finite element spatial discretization of the non-stationary Peterlin viscoelastic fluid model with non-homogeneous boundary conditions. The proposed scheme adds a suitable stabilizing term to improve the structural and stability properties of the scheme. We prove that the scheme is almost unconditionally stable, i.e., stable when the time step is less than or equal to a constant. Further, with the help of the a priori error bounds of the Stokes and Ritz projections, optimal error estimates for the velocity, the conformation tensor and the pressure are presented in suitable norms. Numerical examples are presented that illustrate the accuracy and stability of the scheme. [ABSTRACT FROM AUTHOR]

This paper studies several decoupled penalty methods to overcome the saddle point system of the steady state 2D/3D incompressible magnetohydronamics (MHD). These approaches combine the Oseen iteration and two-level technique with strong uniqueness condition 0 < 2 C 0 2 max { 1 , 2 S c } ‖ F ‖ - 1 (min { R e - 1 , S c C 1 R m - 1 }) 2 ≤ 1 - ‖ F ‖ | - 1 ‖ | F ‖ 0 1 2 < 1 satisfied. For the convenience of implementation, we employ two different simple Lagrange finite element pairs P 1 b - P 1 - P 1 b and P 1 - P 0 - P 1 for velocity field, pressure and magnetic field, respectively. Rigorous analysis of the optimal error estimate and stability are provided. We present comprehensive numerical experiments, which indicate the effectiveness of the proposed methods for both two dimensional and three-dimensional problems. [ABSTRACT FROM AUTHOR]

In this paper we propose a novel way to prescribe weakly the symmetry of stress tensors in weak formulations amenable to the construction of mixed finite element schemes. The approach is first motivated in the context of solid mechanics (using, for illustrative purposes, the linear problem of linear elasticity), and then we apply this technique to reduce the computational cost of augmented fully-mixed methods for thermal convection problems in fluid mechanics, in the case where several additional variables are defined. We show that the new approach allows to maintain the same structure of the mathematical analysis as in the original formulations. Therefore we only need to focus on ellipticity of certain bilinear forms, as this property provides feasible ranges for the stabilization parameters that complete the description of augmented methods. In addition, we present some numerical examples to show that these methods perform better than their counterparts that include vorticity, and emphasize that the reduction in degrees of freedom (and therefore, in computational cost) does not affect the quality of numerical solutions. [ABSTRACT FROM AUTHOR]

The paper is concerned with the unconditional stability and convergence of characteristics type methods for the time-dependent Navier-Stokes equations. We present optimal error estimates in $$L^2$$ and $$H^1$$ norms for a typical modified characteristics finite element method unconditionally, while all previous works require certain time-step restrictions. The analysis is based on an iterated characteristic time-discrete system, with which the error function is split into a temporal error and a spatial error. With a rigorous analysis to the characteristic time-discrete system, we prove that the difference between the numerical solution and the solution of the time-discrete system is $$\tau $$ -independent, where $$\tau $$ denotes the time stepsize. Thus numerical solution in $$W^{1,\infty }$$ is bounded and optimal error estimates can be obtained in a traditional way. Numerical results confirm our analysis and show clearly the unconditional stability and convergence of the modified characteristics finite element method for the time-dependent Navier-Stokes equations. The approach used in this paper can be easily extended to many other characteristics-based methods. [ABSTRACT FROM AUTHOR]

In this paper, we construct a stable finite element pair for incompressible Stokes problem and a discrete Stokes complex associated with that pair on general quadrilateral grids. The finite element spaces involved consist of piecewise polynomials only, and the divergence-free condition is imposed in a primal formulation. These constructions can be generated onto grids that consist of both triangular and quadrilateral cells. [ABSTRACT FROM AUTHOR]

This paper introduces a novel three-field formulation for the Bingham flow problem and its two-dimensional version named after Mosolov together with low-order discretizations: a nonconforming for the classical formulation and a mixed finite element method for the three-field model. The two discretizations are equivalent and quasi-optimal in the sense that the $$H^1$$ error of the primal variable is bounded by the error of the $$L^2$$ best-approximation of the stress variable. This improves the predicted convergence rate by a log factor of the maximal mesh-size in comparison to the first-order conforming finite element method in a model scenario. Despite that numerical experiments lead to comparable results, the nonconforming scheme is proven to be quasi-optimal while this is not guaranteed for the conforming one. [ABSTRACT FROM AUTHOR]