Analysis and finite element approximation of a Ladyzhenskaya model for viscous flow in streamfunction form

In this paper we consider a model for the motion of incompressible viscous flows proposed by Ladyzhenskaya. The Ladyzhenskaya model is written in terms of the velocity and pressure while the studied model is written in terms of the streamfunction only. We derived the streamfunction equation of the Ladyzhenskaya model and present a weak formulation and show that this formulation is equivalent to the velocity–pressure formulation. We also present some existence and uniqueness results for the model. Finite element approximation procedures are presented. The discrete problem is proposed to be well posed and stable. Some error estimates are derived. We consider the 2D driven cavity flow problem and provide graphs which illustrate differences between the approximation procedure presented here and the approximation for the streamfunction form of the Navier–Stokes equations. Streamfunction contours are also displayed showing the main features of the flow.     

Asymptotic expansions and extrapolations of eigenvalues for the stokes problem by mixed finite element methods

This paper derives a general procedure to produce an asymptotic expansion for eigenvalues of the Stokes problem by mixed finite elements. By means of integral expansion technique, the asymptotic error expansions for the approximations of the Stokes eigenvalue problem by Bernadi–Raugel element and _Q2-_P1$Q_2-P_1$ element are given. Based on such expansions, the extrapolation technique is applied to improve the accuracy of the approximations.     

Rate of convergence of regularization procedures and finite element approximations for the total variation flow

Summary We derive rates of convergence for regularization procedures (characterized by a parameter ) and finite element approximations of the total variation flow, which arises from image processing, geometric analysis and materials sciences. Practically useful error estimates, which depend on only in low polynomial orders, are established for the proposed fully discrete finite element approximations. As a result, scaling laws which relate mesh parameters to the regularization parameter are also obtained. Numerical experiments are provided to validate the theoretical results and show efficiency of the proposed numerical methods.     

A reduced finite volume element formulation and numerical simulations based on POD for parabolic problems

A proper orthogonal decomposition (POD) method is applied to a usual finite volume element (FVE) formulation for parabolic equations such that it is reduced to a POD FVE formulation with lower dimensions and high enough accuracy. The error estimates between the reduced POD FVE solution and the usual FVE solution are analyzed. It is shown by numerical examples that the results of numerical computation are consistent with theoretical conclusions. Moreover, it is also shown that the reduced POD FVE formulation based on POD method is both feasible and highly efficient.     

A reduced finite difference scheme based on singular value decomposition and proper orthogonal decomposition for Burgers equation

In this article, a reduced optimizing finite difference scheme (FDS) based on singular value decomposition (SVD) and proper orthogonal decomposition (POD) for Burgers equation is presented. Also the error estimates between the usual finite difference solution and the POD solution of reduced optimizing FDS are analyzed. It is shown by considering the results obtained for numerical simulations of cavity flows that the error between the POD solution of reduced optimizing FDS and the solution of the usual FDS is consistent with theoretical results. Moreover, it is also shown that the reduced optimizing FDS is feasible and efficient.     

Nonconforming mixed finite element approximation to the stationary Navier-Stokes equations on anisotropic meshes

The main aim of this paper is to study the error estimates of a rectangular nonconforming finite element for the stationary Navier-Stokes equations under anisotropic meshes. That is, the nonconforming rectangular element is taken as approximation space for the velocity and the piecewise constant element for the pressure. The convergence analysis is presented and the optimal error estimates both in a broken H¹-norm for the velocity and in an L²-norm for the pressure are derived on anisotropic meshes.     

Continuous interior penalty finite element method for the time-dependent Navier–Stokes equations: space discretization and convergence

This paper focuses on the numerical analysis of a finite element method with stabilization for the unsteady incompressible Navier–Stokes equations. Incompressibility and convective effects are both stabilized adding an interior penalty term giving L ²-control of the jump of the gradient of the approximate solution over the internal faces. Using continuous equal-order finite elements for both velocities and pressures, in a space semi-discretized formulation, we prove convergence of the approximate solution. The error estimates hold irrespective of the Reynolds number, and hence also for the incompressible Euler equations, provided the exact solution is smooth.     

Development of discontinuous Galerkin methods for Maxwell’s equations in metamaterials and perfectly matched layers

The discontinuous Galerkin method has proved to be an accurate and efficient way to numerically solve many differential equations. In this paper, we extend this method to solve the time-dependent Maxwell’s equations when metamaterials and perfectly matched layers are involved. Numerical results are presented to demonstrate that our method is not only simple to implement, but also quite effective in solving Maxwell’s equations in complex media.     

Wick-stochastic finite element solution of reaction–diffusion problems

Real life reaction–diffusion problems are characterized by their inherent or externally induced uncertainties in the design parameters. This paper presents a finite element solution of reaction–diffusion equations of Wick type. Using the Wick-product properties and the Wiener–Itô chaos expansion, the stochastic variational problem is reformulated to a set of deterministic variational problems. To obtain the chaos coefficients in the corresponding deterministic reaction–diffusion, we implement the usual Galerkin finite element method using standard techniques. Once this representation is computed, the statistics of the numerical solution can be easily evaluated. Computational results are shown for one- and two-dimensional test examples.     

An FEM identification procedure for a wing section within a class of finite Laurent series

This paper concerns a determination procedure for conformal mapping of a wing through a finite element computation of potential function associated with the flow of 2-dimensional perfect fluid around the given wing section. Through our numerical procedure a family of mappings is obtained in the forms of finite Laurent series for an initial wing section input. Each member of the family describes a wing section located in a neighboring domain of the input one. Some of them could be expected as modified versions of the original wing section input, although they could not recover completely it.Inputting the shape of wing section has ambiguity in practical cases of wing sections such as the NACA23012 wing section. We would like to postulate that our identification procedure should be employed in the determination process of numerical profiles of the wing section considered, since identified ones are significantly easier in numerical processing than the original input shape.     

Galerkin and subspace decomposition methods in space and time for the Navier-Stokes equations

The Galerkin method and the subspace decomposition method in space and time for the two-dimensional incompressible Navier-Stokes equations with the H²-initial data are considered. The subspace decomposition method consists of splitting the approximate solution as the sum of a low frequency component discretized by the small time step Δt and a high frequency one discretized by the large time step pΔt with p>1. The H²-stability and L²-error analysis for the subspace decomposition method are obtained. Finally, some numerical tests to confirm the theoretical results are provided.     

Microstructure evolution model in micromagnetics

The proposed model combines tendency for minimization of Gibbs magnetic energy with the rate-independent maximum-dissipation mechanism that reflects the macroscopical quantity of energy required to change one pole of a magnet to another. The microstructure is described on a mesoscopical level in terms of Young measures. Such mesoscopical, distributed-parameter model is formulated (and, after a suitable regularization), analyzed, discretized, implemented, and eventually tested computationally on a uni-axial magnet. The desired hysteresis macroscopical response is demonstrated together with the influence of material properties.     

Bubble finite elements for the primitive equations of the ocean

We introduce in this paper two original Mixed methods for the numerical resolution of the (stationary) Primitive Equations (PE) of the Ocean. The PE govern the behavior of oceanic flows in shallow domains for large time scales. We use a reduced formulation (Lions et al. [28]) involving horizontal velocities and surface pressures. By using bubble functions constructed ad-hoc, we are able to define two stable Mixed Methods requiring a low number of degrees of freedom. The first one is based on the addition of bubbles of reduced support to velocities elementwise. The second one makes use of conic bubbles of extended support along the vertical coordinate. The latter constitutes a genuine mini-element for the PE, e.g., it requires the least number of extra degrees of freedom to stabilize piecewise linear hydrostatic pressures. Both methods verify a specific inf-sup condition and provide stability and convergence. Finally, we compare several numerical features of the proposed pairs in the context of other FE methods found in the literature.     

A stabilized finite element method based on two local Gauss integrations for the Stokes equations

This paper considers a stabilized method based on the difference between a consistent and an under-integrated mass matrix of the pressure for the Stokes equations approximated by the lowest equal-order finite element pairs (i.e., the _P1$P_1$–_P1$P_1$ and _Q1$Q_1$–_Q1$Q_1$ pairs). This method only offsets the discrete pressure space by the residual of the simple and symmetry term at element level in order to circumvent the inf–sup condition. Optimal error estimates are obtained by applying the standard Galerkin technique. Finally, the numerical illustrations agree completely with the theoretical expectations.     

Convergence analysis of the FEM approximation of the first order projection method for incompressible flows with and without the inf-sup condition

In this paper we obtain convergence results for the fully discrete projection method for the numerical approximation of the incompressible Navier–Stokes equations using a finite element approximation for the space discretization. We consider two situations. In the first one, the analysis relies on the satisfaction of the inf-sup condition for the velocity-pressure finite element spaces. After that, we study a fully discrete fractional step method using a Poisson equation for the pressure. In this case the velocity-pressure interpolations do not need to accomplish the inf-sup condition and in fact we consider the case in which equal velocity-pressure interpolation is used. Optimal convergence results in time and space have been obtained in both cases.     

Block iterative solvers for higher order finite volume methods

Recently, new higher order finite volume methods (FVM) were introduced in [Z. Cai, J. Douglas, M. Park, Development and analysis of higher order finite volume methods over rectangles for elliptic equations, Adv. Comput. Math. 19 (2003) 3-33], where the linear system derived by the hybridization with Lagrange multiplier satisfying the flux consistency condition is reduced to a linear system for a pressure variable by an appropriate quadrature rule. We study the convergence of an iterative solver for this linear system. The conjugate gradient (CG) method is a natural choice to solve the system, but it seems slow, possibly due to the non-diagonal dominance of the system. In this paper, we propose block iterative methods with a reordering scheme to solve the linear system derived by the higher order FVM and prove their convergence. With a proper ordering, each block subproblem can be solved by fast methods such as the multigrid (MG) method. The numerical experiments show that these block iterative methods are much faster than CG.     

A local superconvergence analysis of the finite element method for the Stokes equations by local projections

In this paper, we focus on a local superconvergence analysis of the finite element method for the Stokes equations by local projections. The local and global superconvergence results of finite element solutions are provided for the Stokes problem under some corresponding regularity assumptions. Conclusion can be drawn that the local superconvergence has advantages over the global superconvergence in two important aspects. On the one hand, it offsets theoretical limitation in practical applications. On the other hand, interior estimates are derived on the base of local properties of the domain without global smoothness for the exact solution and prior regularity of the problem globally over the whole domain.     

An a priori error estimate for a monotone mixed finite-element discretization of a convection–diffusion problem

We present a local exponential fitting hybridized mixed finite-element method for convection–diffusion problem on a bounded domain with mixed Dirichlet Neuman boundary conditions. With a new technique that interpretes the algebraic system after static condensation as a bilinear form acting on certain lifting operators we prove an a priori error estimate on the Lagrange multipliers that requires minimal regularity. While an extension of more classical arguments provide an estimate for the other solution components.     

Numerical analysis of electric field formulations of the eddy current model

This paper deals with finite element methods for the numerical solution of the eddy current problem in a bounded conducting domain crossed by an electric current, subjected to boundary conditions involving only data easily available in applications. Two different cases are considered depending on the boundary data: input current intensities or differences of potential. Weak formulations in terms of the electric field are given in both cases. In the first one, the input current intensities are imposed by means of integrals on curves lying on the boundary of the domain and joining current entrances and exit. In the second one, the electric potentials are imposed by means of Lagrange multipliers, which are proved to represent the input current intensities. Optimal error estimates are proved in both cases and implementation issues are discussed. Finally, numerical tests confirming the theoretical results are reported. Partially supported by Xunta de Galicia research grant PGIDIT02PXIC20701PN (Spain). Partially supported by FONDAP in Applied Mathematics (Chile). Partially supported by MCYT (Spain) project DPI2003-01316.