Error Estimates For Velocity Research Articles

Superconvergence error analysis of linearized semi-implicit bilinear-constant SAV finite element method for the time-dependent Navier–Stokes equations

In this paper, based on the scalar auxiliary variable (SAV) approach, the superconvergence error analysis is investigated for the time-dependent Navier–Stokes equations. In which, an equivalent system of the Navier–Stokes equations with three variables and a fully-discrete scheme is developed with semi-implicit Euler discretization for the temporal direction and low-order bilinear-constant finite element discretization for the spatial direction, respectively. With the help of the high-precision estimations of the bilinear-constant finite element pair on the rectangular meshes, the superclose error estimates for velocity in H1-norm and pressure in L2-norm are obtained by treating the trilinear term carefully and skillfully. The global superconvergence results are also derived in terms of a simple and efficient interpolation post-processing technique. Finally, some numerical results are provided to demonstrate the correctness of the theoretical analysis.

Superconvergence analysis of the bilinear‐constant scheme for two‐dimensional incompressible convective Brinkman–Forchheimer equations

AbstractIn this article, a low order conforming mixed finite element method is proposed and investigated for two‐dimensional convective Brinkman–Forchheimer equations. Based on the special properties of the bilinear‐constant finite element pair on the rectangular mesh and the careful treatment of the nonlinear terms, the superclose error estimates for velocity in ‐norm and pressure in ‐norm are obtained. Then, in terms of interpolation post‐processing technique, the global superconvergence results are derived. Finally, some numerical experiments are carried out to demonstrate the correctness of the theoretical findings.

Unconditional Superconvergence Error Estimates of Semi-Implicit Low-Order Conforming Mixed Finite Element Method for Time-Dependent Navier–Stokes Equations

In this paper, the unconditional superconvergence error analysis of the semi-implicit Euler scheme with low-order conforming mixed finite element discretization is investigated for time-dependent Navier–Stokes equations. In terms of the high-accuracy error estimates of the low-order finite element pair on the rectangular mesh and the unconditional boundedness of the numerical solution in L∞-norm, the superclose error estimates for velocity in H1-norm and pressure in L2-norm are derived firstly by dealing with the trilinear term carefully and skillfully. Then, the global superconvergence results are obtained with the aid of the interpolation post-processing technique. Finally, some numerical experiments are carried out to support the theoretical findings.

Electric potential-robust iterative analysis of charge-conservative conforming FEM for thermally coupled inductionless MHD system

This paper mainly considers electric potential-robust iterative methods based on conservation of charge in Lipschitz domain for the 2D/3D stationary thermally coupled inductionless MHD equations. Based on the hybrid finite element method, the unknowns of hydrodynamic are discretized by the stable velocity–pressure finite element pair, and the current density along with electric potential are similarly discretized by the conforming finite element pair in H(div,Ω)×L2(Ω). It is proved especially that the optimal error estimates of velocity, current density, temperature and pressure do not depend on electric potential. And on account of the strong nonlinearity of the equations, we present three coupled iterative methods, namely, the Stokes, Newton and Oseen iterations and the convergence and stability under different uniqueness conditions are analyzed strictly. The theoretical analysis is validated by the given numerical results, and for the proposed methods, the applicability and effectiveness are demonstrated.

A decoupled, unconditionally energy stable and charge-conservative finite element method for inductionless magnetohydrodynamic equations

In this paper, we propose a decoupled, unconditionally energy stable and charge-conservative finite element method for the inductionless magnetohydrodynamic equations. The time marching is combined with a first order semi-implicit Euler scheme, a first order perturbation term and some delicate implicit-explicit treatments for coupling terms. The finite element discretization is based on mixed conforming elements, where the hydrodynamic unknowns are approximated by stable finite element pairs, and the current density and electric potential are discretized by divergence-conforming face elements and discontinuous volume elements. This fully discrete scheme only needs to solve a linear subsystem at each time step and yields an exactly divergence-free current density directly. We show that the proposed scheme is well-posed and unconditionally energy stable. Under a weak regularity hypothesis on the exact solution, we present the error estimates for velocity, current density and electric potential. Finally, the validity, reliability, and accuracy of the proposed algorithm are supported by several numerical experiments.

Error analysis of a fully discrete consistent splitting MAC scheme for time dependent Stokes equations

We construct and analyze a numerical scheme based on the truly consistent splitting approach in time and the MAC discretization in space for the time dependent Stokes equations. The scheme only requires solving several Poisson type equations for the velocity and pressure at each time step. We establish the equivalence between two different formulations of the fully discrete consistent splitting schemes, prove unconditional stability, and establish first-order in time and second-order in space error estimates for velocity and pressure in different discrete norms. We also provide numerical experiments to verify our theoretical results.

Error analysis of a conservative finite element scheme for time-dependent inductionless MHD problem

In this paper, we develop and analyze a fully discrete mixed finite element method for unsteady inductionless magnetohydrodynamics problem. An Euler semi-implicit scheme is proposed with a mixed variational formulation based on the variables (u,p,j,ϕ), in which the Navier–Stokes equations are approximated by stable finite elements and the current density is discretized by the divergence-conforming finite element. This scheme has the feature that the discrete current density keeps charge conservation property. It is shown that both the continuous problem and its fully discrete Euler semi-implicit scheme are well-posed. We prove unconditionally convergence and error estimates for velocity, pressure and current density. Finally, numerical experiments have been performed to validate the theoretical analysis and the law of charge conservation.

A Mixed Finite Element Method for Stationary Magneto-Heat Coupling System with Variable Coefficients.

In this article, a mixed finite element method for thermally coupled, stationary incompressible MHD problems with physical parameters dependent on temperature in the Lipschitz domain is considered. Due to the variable coefficients of the MHD model, the nonlinearity of the system is increased. A stationary discrete scheme based on the coefficients dependent temperature is proposed, in which the magnetic equation is approximated by Nédélec edge elements, and the thermal and Navier–Stokes equations are approximated by the mixed finite elements. We rigorously establish the optimal error estimates for velocity, pressure, temperature, magnetic induction and Lagrange multiplier with the hypothesis of a low regularity for the exact solution. Finally, a numerical experiment is provided to illustrate the performance and convergence rates of our numerical scheme.

A family of quadratic finite volume method for solving the Stokes equation

This paper presents a family of stable quadratic finite volume methods for solving the Stokes equation over triangular meshes. We introduce a novel mapping, which connects trial spaces with test spaces of the finite volume methods, and impose a geometric condition on the triangular meshes of primary partitions to ensure the ellipticity of discrete elliptic bilinear forms in the finite volume methods. By using the abstract theories of generalized saddle-point problems, the stability (or inf-sup condition in general) of the proposed finite volume methods could be established. The resulting finite volume methods have the optimal H1-norm error estimate for velocity of the Stokes problem without constraints on dual meshes of dual partitions of the finite volume methods, and have the optimal L2-norm error estimate for velocity with a specially selected dual mesh of the dual partitions. In addition, by adopting another appropriate dual mesh of the dual partitions, the finite volume methods may be identified as the corresponding finite element methods with differences only in their right-hand items. As a result, the optimal-order error estimate could be obtained in the H1-norm for velocity without any restriction on mesh geometry of the primary partitions. Finally, we investigate numerically the imposed geometric conditions on meshes of the primary partitions for ensuring the ellipticity of the discrete elliptic bilinear forms and we validate our theoretical error estimates through some numerical experiments.

Superconvergence of a WG method for the Stokes equations with continuous pressure

In this paper, a new stabilizer free weak Galerkin finite element method with continuous finite element approximation of the pressure is proposed and analyzed for the Stokes problem. To achieve the same accuracy of the pressure approximation, the new method has less degrees of freedom among all existing weak Galerkin methods for the Stokes problem. Another feature of the new method is the superclose property for the approximation of the velocity. Error estimates for velocity in both the energy norm and L2 norm and the estimate for pressure in L2 norm are established. A local postprocessing technique is presented to obtain global superconvergence result. Numerical examples are shown to verify the theoretical results.

Pressure robust hybrid mixed finite element method for Darcy–Forchheimer model

In this paper, we propose a pressure robust hybrid mixed finite element method for Darcy–Forchheimer model. The method uses piecewise polynomials of degrees for the velocity and k−1 for pressure in the interior of elements inside the domain, and piecewise polynomials of degree k for the numerical traces of the pressure on the inter-element boundaries. Our method can get the globally divergence-free approximation solution of velocity. Optimal error estimates are derived and the error estimates of velocity are pressure robust. Finally, numerical experiments confirm the theoretical results.

A Mixed Finite-Element Method on Polytopal Mesh

In this paper, we introduce new stable mixed finite elements of any order on polytopal mesh for solving second-order elliptic problem. We establish optimal order error estimates for velocity and super convergence for pressure. Numerical experiments are conducted for our mixed elements of different orders on 2D and 3D spaces that confirm the theory.

Numerical analysis of fully discrete finite element methods for the stochastic Navier-Stokes equations with multiplicative noise

Previous work on the stability and convergence analysis of the finite element methods for the deterministic Navier-Stokes equations was carried out under the uniqueness condition. In this paper, the corresponding results of fully discrete finite element method are developed for the stochastic Navier-Stokes equations with multiplicative noise. The stability and error estimates for velocity in expectation are rigorously proved, and the L2-norm in a time-averaged version for pressure is provided with convergence rates in spatial and temporal form O(Δt+hΔt). Finally, numerical experiments are given to illustrate the features of the proposed numerical method and verify the theoretical claims.

Barycentric Rational Collocation Method for the Incompressible Forchheimer Flow in Porous Media

Barycentric rational collocation method is introduced to solve the Forchheimer law modeling incompressible fluids in porous media. The unknown velocity and pressure are approximated by the barycentric rational function. The main advantages of this method are high precision and efficiency. At the same time, the algorithm and program can be expanded to other problems. The numerical stability can be guaranteed. The matrix form of the collocation method is obtained from the discrete numerical schemes. Numerical analysis and error estimates for velocity and pressure are established. Numerical experiments are carried out to validate the convergence rates and show the efficiency.

A Uniformly Robust Staggered DG Method for the Unsteady Darcy-Forchheimer-Brinkman Problem

In this paper, we propose and analyze a uniformly robust staggered DG method for the unsteady Darcy-Forchheimer-Brinkman problem. Our formulation is based on velocity gradient-velocity-pressure and the resulting scheme can be flexibly applied to fairly general polygonal meshes. We relax the tangential continuity for velocity, which is the key ingredient in achieving the uniform robustness. We present well-posedness and error analysis for both the semi-discrete scheme and the fully discrete scheme, and the theories indicate that the error estimates for velocity are independent of pressure. Several numerical experiments are presented to confirm the theoretical findings.

A divergence-free reconstruction of the nonconforming virtual element method for the Stokes problem

In this paper, we propose and investigate a divergence-free reconstruction of the nonconforming virtual element for the Stokes problem. By constructing the computable Raviart–Thomas-like interpolation operator, we guarantee the independence between the velocity error estimation |u−uh|1,h and the continuous pressure p, as it happens for the divergence-free flow solver. Moreover, this modified scheme can also inherit the advantages of the classical nonconforming virtual element method, such as, very general meshes including non-convex and degenerate elements, a unified scheme for an arbitrary-order approximation accuracy k, etc. Then, we provide the optimal L2-error estimates for the velocity gradient and the pressure by taking advantage of the Raviart–Thomas-like interpolation operator and avoiding the use of a trace inequality. Finally, three numerical experiments are presented to conform the theoretical analysis.

Numerical analysis of the unconditionally stable discontinuous Galerkin schemes for the nonstationary conduction–convection problem

This work aims at employing the discontinuous Galerkin (DG) methods for solving the nonstationary conduction–convection model, which is motivated by advantages of DG methods. The discontinuous element pair Pk-Pk−1-Pk in the space discretization and backward Euler scheme in time are applied, as well as an explicit upwind scheme based on Picard iteration is introduced for the coupled term. Existence and uniqueness of the approximation solutions are proved, along with preserving the unconditional stability of the numerical schemes. The priori error estimates for velocity, pressure and temperature are also derived. Numerical tests are provided to verify the theoretical analysis and to indicate the efficiency of the DG methods in simulating practical problems.

Fully discrete finite element approximation of the 2D/3D unsteady incompressible magnetohydrodynamic-Voigt regularization flows

We devote the present paper to a fully discrete finite element scheme for the 2D/3D nonstationary incompressible magnetohydrodynamic-Voigt regularization model. This scheme is based on a finite element approximation for space discretization and the Crank-Nicolson-type scheme for time discretization, which is a two-step method. Moreover, we study stability and convergence of the fully discrete finite element scheme and obtain unconditional stability and error estimates of velocity and magnetic fields, respectively. Finally, several numerical experiments are investigated to confirm our theoretical findings.

In-Motion Alignment for Doppler Velocity Log-Aided SINS Based on Initial Velocity Error Estimation

Inertial frame alignment (IFA) for strapdown inertial navigation systems is widely considered to be a promising method due to its robustness to disturbances and noise. By using the velocity data from a Doppler velocity log (DVL), the IFA can be performed while the vehicle is in motion. However, this article reveals that the initial velocity error of the DVL at the startup continuously and considerably affects the precision of the in-motion IFA. Based on the formula derived with other sensor errors isolated, the azimuth error is proportional to the initial velocity error and is inversely proportional to the square of time, e.g., a 0.3 m/s measurement error in the initial velocity from the Doppler velocity log leading to a considerable azimuth error of more than 1.78° in five minutes. Moreover, the azimuth error becomes bulky if the latitude is high. The modified cost function with a constant vector is used to replace the cost function from the well-known Wahba’s problem. Thus, an alignment result with a much higher precision and with the initial velocity error suppressed can be obtained in a short period of time. The vehicular experimental results further verify the error analysis and the improved method.

Performance Enhancement for Conventional Tightly Coupled INS/DVL Navigation System Using Regeneration of Partial DVL Measurements

Inertial navigation systems/Doppler velocity log (INS/DVL) integrated navigation systems are widely used in underwater environments where GPS is unavailable. An INS/DVL integrated navigation system is generally loosely coupled; however, this does not work if any of the DVL transducers do not work. If a system is tightly coupled, velocity error can be estimated with fair accuracy even if some of the transducers fail. However, despite the robustness of a tightly coupled system compared to a loosely coupled one, velocity error estimation accuracy of the former decreases as the number of faulty transducers increases. Therefore, this paper proposes an INS/DVL/revolutions per minute (RPM) integrated navigation filter designed to improve the performance of conventional tightly coupled integrated systems by estimating data from faulty transducers using RPM data. Two salient features of the proposed filter are (1) estimating RPM data accounting for error from the effect of tidal currents and (2) continuous estimation of error in RPM data by selectively converting only the measurements of faulty transducers. The performance of the proposed filter was first verified using Monte Carlo numerical simulations with the analysis range set to 1 standard deviation (1σ, 68%) and then with real sea test measurement data.