Generalized Darcy–Oseen resolvent problem

In this paper, we study the well‐posedness of a coupled Darcy–Oseen resolvent problem, describing the fluid flow between free‐fluid domains and porous media separated by a semipermeable membrane. The influence of osmotic effects, induced by the presence of a semipermeable membrane, on the flow velocity is reflected in the transmission conditions on the surface between the free‐fluid domain and the porous medium. To prove the existence of a weak solution of the generalized Darcy–Oseen resolvent system, we consider two auxiliary problems: a mixed Navier–Dirichlet problem for the generalized Oseen resolvent system and Robin problem for an elliptic equation related to the general Darcy equations. © 2016 The Authors Mathematical Methods in the Applied Sciences Published by John Wiley & Sons Ltd.     

Stabilized finite element schemes for generalized Oseen systems

In recent works [1, 2], advanced approximation schemes for the numerical calculation of compressible viscous flow were developed, analyzed theoretically and applied successfully to benchmark problems. These methods are based on splitting the Poisson–Stokes system (1.1) describing the motion of a viscous compressible fluid into a generalized Oseen problem for the velocity and a hyperbolic transport equation for the density. Highly refined finite element techniques were proposed for the numerical solution of these separated subproblems of simpler structure; cf. [2]. In this paper, error estimates for a SUPG/(PSPG) and grad-div stabilized finite element approximation of the generalized Oseen problems that arise in the course of the splitting procedure are presented. LBB-stable pairs of finite element spaces are used for velocity and pressure. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A C finite element method for an inverse problem

A C⁰ finite element method is presented for an inverse problem in which the coefficient in the differential operator is to be determined from the measurement of the solution of a boundary value problem. The unknown in the inverse problem is approximated by a minimizer of a cost function that includes both the output error and equation error. Error estimates in a weighted H⁻¹ norm and L² are given. Numerical examples are presented to show features of the method.     

Two-sided a posteriori estimates for a generalized Stokes problem

The paper is concerned with a three-field statement of a generalized Stokes problem related to viscous flow problems for fluids with polymeric chains. For homogeneous Dirichlét boundary conditions, this model and respective numerical methods have been studied previously. In the present paper, a generalized Stokes problem with variable viscosity and nonhomogeneous Dirichlét or mixed Dirichlét/Neumann boundary conditions is considered, and functional a posteriori error estimates for the velocity, pressure, and stress fields are derived. The estimates are practically computable, sharp (i.e., have no gap between the left- and right-hand sides), and are valid for arbitrary functions from respective functional classes. The estimates are obtained by transformations of the integral identity that assigns the generalized solution (this method was suggested and used earlier for certain classes of elliptic type problems). Error majorants are weighted sums of terms penalizing violations of the constitutive, equilibrium, and divergence relations with weights determined by the constants in the Friederichs inequality and the inf-sup (LBB) condition. Bibliography: 53 titles. Dedicated to the jubilee of Professor V. A. Solonnikov Published in Zapiski Nauchnykh Seminarov POMI, Vol. 362, 2008, pp. 272–302.     

Improved stability and error analysis for a class of local projection stabilizations applied to the Oseen problem

We consider a class of local projection stabilizations with projection spaces defined on (possibly) overlapping sets applied to the Oseen problem. We prove that the underlying bilinear form satisfies an inf–sup condition with respect to a stronger norm than coercivity suggests. A modification of the stabilization of the convection allows an optimal estimation of the consistency error. A priori estimates in the stronger norm and in the L² norm for the pressure are established. Discontinuous pressure approximations are included in the analysis. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2013     

Weighted Clément operator and application to the finite element discretization of the axisymmetric Stokes problem

We consider the Stokes problem in an axisymmetric three-dimensional domain with data which are axisymmetric and have angular component equal to zero. We observe that the solution is also axisymmetric and the velocity has also zero angular component, hence the solution satisfies a system of equations in the meridian domain. The weak three-dimensional problem reduces to a two-dimensional one with weighted integrals. The latter is discretized by Taylor–Hood type finite elements. A weighted Clément operator is defined and approximation results are proved. This operator is then used to derive the discrete inf–sup condition and optimal a priori error estimates.     

The regularization method for an obstacle problem

Summary. We give a relatively complete analysis for the regularization method, which is usually used in solving non-differentiable minimization problems. The model problem considered in the paper is an obstacle problem. In addition to the usual convergence result and a-priori error estimates, we provide a-posteriori error estimates which are highly desired for practical implementation of the regularization method. Received March 22, 1993 / Revised version received October 11, 1993     

Estimates of a solution of the model,evolution, generalized stokes problem in weighted Hölder spaces

We prove coercive estimates in anisotropic weighted Hölder spaces for solution of the model Cauchy-Dirichlet problem in the half-space for a generalized Stokes system. Bibliography: 8 titles.     

A weighted Lq‐approach to Oseen flow around a rotating body

We study time-periodic Oseen flows past a rotating body in ℝ³ proving weighted a priori estimates in L^q-spaces using Muckenhoupt weights. After a time-dependent change of coordinates the problem is reduced to a stationary Oseen equation with the additional terms (ω ∧ x) ⋅ ∇ u and −ω ∧ u in the equation of momentum where ω denotes the angular velocity. Due to the asymmetry of Oseen flow and to describe its wake we use anisotropic Muckenhoupt weights, a weighted theory of Littlewood–Paley decomposition and of maximal operators as well as one-sided univariate weights, one-sided maximal operators and a new version of Jones' factorization theorem. Copyright © 2007 John Wiley & Sons, Ltd.     

The Robin problem for the scalar Oseen equation

We study the Robin problem for the scalar Oseen equation in an open n‐dimensional set with compact Ljapunov boundary. We prescribe two types of Robin boundary conditions, and prove the unique solvability of these problems as well as a representation formula for the solution in form of a scalar Oseen single layer potential. Moreover, we prove the maximum principle for the solution to the Robin problem of the scalar Oseen equation. Copyright © 2013 John Wiley & Sons, Ltd.     

Local and pointwise error estimates of the local discontinuous Galerkin method applied to the Stokes problem

We prove local and pointwise error estimates for the local discontinuous Galerkin method applied to the Stokes problem in two and three dimensions. By using techniques originally developed by A. Schatz [Math. Comp., 67 (1998), 877-899] to prove pointwise estimates for the Laplace equation, we prove optimal weighted pointwise estimates for both the velocity and the pressure for domains with smooth boundaries.

     

Defect correction method for viscoelastic fluid flows at high Weissenberg number

We study a defect correction method for the approximation of viscoelastic fluid flow. In the defect step, the constitutive equation is computed with an artificially reduced Weissenberg parameter for stability, and the resulting residual is corrected in the correction step. We prove the convergence of the defect correction method and derive an error estimate for the Oseen‐viscoelastic model problem. The derived theoretical results are supported by numerical tests for both the Oseen‐viscoelastic problem and the Johnson‐Segalman model problem. © 2005 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2006     

Method of penalization for the state equation for an elliptical optimal control problem

We solve by finite difference method an optimal control problem of a system governed by a linear elliptic equation with pointwise control constraints and non-local state constraints. A discrete optimal control problem is approximated by a minimization problem with penalized state equation. We derive the error estimates for the distance between the exact and regularized solutions. We also prove the rate of convergence of block Gauss–Seidel iterative solution method for the penalized problem. We present and analyze the results of the numerical experiments.     

Efficient and reliable hierarchical error estimates for an elliptic obstacle problem

We present and analyze novel hierarchical a posteriori error estimates for a self-adjoint elliptic obstacle problem. Under a suitable saturation assumption, we prove the efficiency and reliability of our hierarchical estimates. The proof is based upon some new observations on the efficiency of some hierarchical error indicators. These new observations allow us to remove an additional regularity condition on the underlying grid required in the previous analysis. Numerical computations confirm our theoretical findings.     

A posteriori error estimates for the generalized Stokes problem

The paper concerns a posteriori estimates of functional type for the difference between exact and approximate solutions to a generalized Stokes problem. The estimates are derived by transformations of the basic integral identity defining a generalized solution to the problem using the method suggested by the first author. The estimates obtained can be classified into two types. Estimates of the first type are valid only for solenoidal functions, while estimates of the second type are applicable for any functions that belong to the energy space of the respective problem and satisfy the boundary conditions. In the second case, the estimates include an additional penalty term with a multiplier defined by the constant in the Ladyzhenskaya-Babuška-Brezzi condition. It is proved that a posteriori estimates for the velocity field yield computable estimates of the difference between exact and approximate pressure functions in the L₂-norm. It is shown that the estimates provide sharp upper and lower bounds of the error and their practical computation requires to solve only finite-dimensional problems. Bibliography: 34 titles. __________ Translated from Problemy Matematicheskogo Analiza, No. 34, 2006, pp. 89–101.     

A posteriori error estimates for a finite element discretization of interior point methods for an elliptic optimization problem with state constraints

In this paper we are concerned with a posteriori error estimates for the solution of some state constraint optimization problem subject to an elliptic PDE. The solution is obtained using an interior point method combined with a finite element method for the discretization of the problem. We will derive separate estimates for the error in the cost functional introduced by the interior point parameter and by the discretization of the problem. Finally we show numerical examples to illustrate the findings for pointwise state constraints and pointwise constraints on the gradient of the state.     

‐solution of the Neumann,Robin and transmission problem for the scalar Oseen equation

We find necessary and sufficient conditions for the existence of an ‐solution of the Neumann problem, the Robin problem and the transmission problem for the scalar Oseen equation in three‐dimensional open sets. As a consequence we study solutions of the generalized jump problem.     

A generalized quasi-boundary value method for the backward semi-linear time-fractional heat problem in a cylinder

We are concerned with a backward problem associated with a semi-linear time-fractional heat equation in an axis-symmetric cylinder, which arises from the modeling of the blast furnace steelmaking in metallurgy. Under some assumptions, the existence and uniqueness of the solution to the semi-linear problem is first established. The ill-posedness of the backward problem is then established, and we obtain the error estimates by a generalized quasi-boundary value regularization method. Finally, the numerical experiment is presented to demonstrate the effectiveness of the proposed method.     

On Two Iterative Least-Squares Finite Element Schemes for the Incompressible Navier–Stokes Problem

This paper is mainly devoted to a comparative study of two iterative least-squares finite element schemes for solving the stationary incompressible Navier–Stokes equations with velocity boundary condition. Introducing vorticity as an additional unknown variable, we recast the Navier–Stokes problem into a first-order quasilinear velocity–vorticity–pressure system. Two Picard-type iterative least-squares finite element schemes are proposed to approximate the solution to the nonlinear first-order problem. In each iteration, we adopt the usual L ² least-squares scheme or a weighted L ² least-squares scheme to solve the corresponding Oseen problem and provide error estimates. We concentrate on two-dimensional model problems using continuous piecewise polynomial finite elements on uniform meshes for both iterative least-squares schemes. Numerical evidences show that the iterative L ² least-squares scheme is somewhat suitable for low Reynolds number flow problems, whereas for flows with relatively higher Reynolds numbers the iterative weighted L ² least-squares scheme seems to be better than the iterative L ² least-squares scheme. Numerical simulations of the two-dimensional driven cavity flow are presented to demonstrate the effectiveness of the iterative least-squares finite element approach.     

Two‐grid variational multiscale algorithms for the stationary incompressible Navier‐Stokes equations with friction boundary conditions

Two‐grid variational multiscale (VMS) algorithms for the incompressible Navier‐Stokes equations with friction boundary conditions are presented in this article. First, one‐grid VMS algorithm is used to solve this problem and some error estimates are derived. Then, two‐grid VMS algorithms are proposed and analyzed. The algorithms consist of nonlinear problem on coarse grid and linearized problem (Stokes problem or Oseen problem) on fine grid. Moreover, the stability and convergence of the present algorithms are established. Finally, Numerical results are shown to confirm the theoretical analysis. © 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 546–569, 2017