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Abstract: The purpose of this paper is to present the SUPG finite element method for adiabatic flows and to compare the results with those obtained via various computational methods. The incompressibility assumption is often used to solve transient viscous flows. On the other hand, compressible flow analyses are also popular because natural flows include compressibility even if it is a negligible amount. It is well known that incompressibility is a limit state of compressibility. To solve compressible flows, three kinds of governing equations are needed: the conservation of mass, momentum, and energy, in which density, velocity, and internal energy are independent variables. If we assume that there is no heat transfer in or out of a system, the energy equation can be eliminated from the governing equations. Density and velocity can be considered independent variables. These kinds of flows are referred to as adiabatic flows in this paper. The SUPG formulation is one of the most widely used methods in the finite element analyses of fluid flows. In this paper, the SUPG method for adiabatic flows is presented. The polytropic law is introduced for the equation of state. Moreover, the computational results obtained by four models of fluid flows are compared: adiabatic flows, compressible flows, constant acoustic velocity flows, and incompressible flows. Verifications are carried out using a lid-driven cavity flow. Boundary effects are estimated using a wide computational domain. Drag forces are compared with those computed using incompressible flows. The pressure coefficients over a surface of a circular cylinder located in fluid flows computed by the present method show good correspondence with the experimental measurement. On the other hand, significant discrepancies are observed between the pressure coefficients computed assuming constant density and the experimental results. [Copyright &y& Elsevier]

In this paper, stabilized continuous finite element methods are analyzed for numerically solving the flux which may be a non H 1 solution. Coercivity and error estimates are established. Numerical experiments are performed to illustrate these methods. [ABSTRACT FROM AUTHOR]

We propose an approach for goal-oriented error estimation in finite element approximations of second-order elliptic problems that combines the dual-weighted residual method and equilibrated-flux reconstruction methods for the primal and dual problems. The objective is to be able to consider discretization schemes for the dual solution that may be different from those used for the primal solution. It is only assumed here that the discretization methods come with a priori error estimates and an equilibrated-flux reconstruction algorithm. A high-order discontinuous Galerkin (dG) method is actually the preferred choice for the approximation of the dual solution thanks to its flexibility and straightforward construction of equilibrated fluxes. One contribution of the paper is to show how the order of the dG method for asymptotic exactness of the proposed estimator can be chosen in the cases where a conforming finite element method, a dG method, or a mixed Raviart–Thomas method is used for the solution of the primal problem. Numerical experiments are also presented to illustrate the performance and convergence of the error estimates in quantities of interest with respect to the mesh size. [ABSTRACT FROM AUTHOR]

The aim of this paper is to provide a-priori error estimates for problems involving curved interfaces and solved with the linear or quadratic extended finite-element method (X-FEM), with particular emphasis on the influence of the geometry representation and the quadrature. We focus on strong discontinuity problems, which covers the case of holes in a material or cracks not subjected to contact as the main applications. The well-known approximation of the curved geometry based on the interpolated level-set function and straight linear or curved quadratic subcells is used, whose accuracy is quantified by means of an appropriate error measure. A priori error estimates are then derived, which depend upon the interpolation order of the displacement, and foremost upon the above error measure and the quadrature scheme in the subcells. The theoretical predictions are successfully compared with numerical experiments. [ABSTRACT FROM AUTHOR]

Abstract: This paper presents an overview of an efficient FEM solver with an accuracy control strategy using automatic local grid refinement. Cheap accuracy estimators are briefly described and some concepts concerning a multigrid FE solver are presented. Several (2D/3D) numerical examples demonstrate the efficiency of the proposed strategy in terms of computation time and memory requirement. Some large scale calculations are presented, up to 200 million degrees of freedom. A particular implementation care has been devoted to ensure both a linear time complexity of every intermediate step and a minimal memory requirement. [Copyright &y& Elsevier]

Abstract: Dirac-δ distributions are often crucial components of the solid–fluid coupling operators in immersed solution methods for fluid–structure interaction (FSI) problems. This is certainly so for methods like the immersed boundary method (IBM) or the immersed finite element method (IFEM), where Dirac-δ distributions are approximated via smooth functions. By contrast, a truly variational formulation of immersed methods does not require the use of Dirac-δ distributions, either formally or practically. This has been shown in the finite element immersed boundary method (FEIBM), where the variational structure of the problem is exploited to avoid Dirac-δ distributions at both the continuous and the discrete level. In this paper, we generalize the FEIBM to the case where an incompressible Newtonian fluid interacts with a general hyperelastic solid. Specifically, we allow (i) the mass density to be different in the solid and the fluid, (ii) the solid to be either viscoelastic of differential type or purely elastic, and (iii) the solid to be either compressible or incompressible. At the continuous level, our variational formulation combines the natural stability estimates of the fluid and elasticity problems. In immersed methods, such stability estimates do not transfer to the discrete level automatically due to the non-matching nature of the finite dimensional spaces involved in the discretization. After presenting our general mathematical framework for the solution of FSI problems, we focus in detail on the construction of natural interpolation operators between the fluid and the solid discrete spaces, which guarantee semi-discrete stability estimates and strong consistency of our spatial discretization. [Copyright &y& Elsevier]

Abstract: In this paper, we present a posteriori error analysis for finite element approximation of convex optimal control problems. We derive a new quasi-interpolation operator of Clément type and a new quasi-interpolation operator of Scott–Zhang type that preserves homogeneous boundary condition. The Scott–Zhang type quasi-interpolation is suitable for an application in bounding the errors in -norm. Then a posteriori error estimators are obtained for the coupled state and control approximations. Such estimators can be used to construct reliable adaptive finite elements for the control problems. [Copyright &y& Elsevier]

Abstract: In this paper we study a new local stabilized nonconforming finite element method based on two local Gauss integrals for solving the stationary Navier–Stokes equations. This nonconforming method utilizes the lowest equal-order pair of mixed finite elements (i.e., ). Error estimates of optimal order are obtained, and numerical results agreeing with these estimates are demonstrated. Numerical comparisons with other mixed finite element methods for solving the Navier–Stokes equations are also presented to show the better performance of the present method. [ABSTRACT FROM AUTHOR]

Abstract: This paper presents a robust a posteriori residual error estimator for diffusion–convection–reaction problems with anisotropic diffusion, approximated by a SUPG finite element method on isotropic or anisotropic meshes in , or 3. The equivalence between the energy norm of the error and the residual error estimator is proved. Numerical tests confirm the theoretical results. [ABSTRACT FROM AUTHOR]

Abstract: This paper presents a strategy for a posteriori error estimation for substructured problems solved by non-overlapping domain decomposition methods. We focus on global estimates of the discretization error obtained through the error in constitutive relation for linear mechanical problems. Our method allows to compute error estimate in a fully parallel way for both primal (BDD) and dual (FETI) approaches of non-overlapping domain decomposition whatever the state (converged or not) of the associated iterative solver. Results obtained on an academic problem show that the strategy we propose is efficient in the sense that correct estimation is obtained with fully parallel computations; they also indicate that the estimation of the discretization error reaches sufficient precision in very few iterations of the domain decomposition solver, which enables to consider highly effective adaptive computational strategies. [ABSTRACT FROM AUTHOR]