Superconvergence for a mixed finite elmenent method for elastic wave propagation in a plane domain

Summary A higher order mixed finite element method is introduced to approximate the solution of wave propagation in a plane elastic medium. A quasi-projection analysis is given to obtain error estimates in Sobolev spaces of nonpositive index. Estimates are given for difference quotients for a spatially periodic problem and superconvergence results of the same type as those of Bramble and Schatz for Galerkin methods are derived.     

A nonoverlapping domain decomposition method for stabilized finite element approximations of the Oseen equations

A nonoverlapping domain decomposition algorithm of Robin–Robin type is applied to the discretized Oseen equations using stabilized finite element approximations of velocity and pressure thus allowing in particular equal-order interpolation. As a crucial result we have to inspect the proof of a modified inf–sup condition, in particular, the dependence of the stability constant with respect to the Reynolds number (cf. appendix). After proving coercivity and strong convergence of the method, we derive an a posteriori estimate which controls convergence of the discrete subdomain solutions to the global discrete solution provided that jumps of the discrete solution converge at the interface. Furthermore, we obtain information on the design of some free parameters within the Robin-type interface condition which essentially influence the convergence speed. Some numerical results confirm the theoretical ones.     

Domain decomposition and splitting methods for Mortar mixed finite element approximations to parabolic equations

Summary We introduce in this article a new domain decomposition algorithm for parabolic problems that combines Mortar Mixed Finite Element methods for the space discretization with operator splitting schemes for the time discretization. The main advantage of this method is to be fully parallel. The algorithm is proven to be unconditionally stable and a convergence result in (Δt/h ^1/2) is presented.     

Augmented hybrid finite element method for domain decomposition

We present an Augmented Hybrid Finite Element Method for Domain Decompositon. In this method, finite element approximations are defined independently on each subdomain and do not match at interface. This dows the user to mda local change of design, of meshes on one aubdomain without modifying other subdomains. Optimal reaults are obtained for a second-order model problem.     

A mixed finite element method for the Stokes equations

A new mixed finite element for the Stokes equations is considered. This new finite element is based on a mixed formulation of the Stokes problem in which the gradient of the velocity is introduced and the velocity is approximated by the Raviart-Thomas element [1]. Optimal error estimates are derived. The number of degrees of freedom, for this element, is the lowest possible, and the local conservation of the mass is assured. A hybrid version of the mixed method is also considered. Finally, some numerical results for the incompressible Navier-Stokes equations are presented. © 1994 John Wiley & Sons, Inc.     

An analysis of a mixed finite element method for the Navier-Stokes equations

Summary A simple mixed finite element method is developed to solve the steady state, incompressible Navier-Stokes equations in a neighborhood of an isolated—but not necessarily unique—solution. Convergence is established under very mild restrictions on the triangulation, and, when the solution is sufficiently smooth, optimal error bounds are obtained.     

B-spline based finite element method in one-dimensional discontinuous elastic wave propagation

The B-spline variant of the finite element method (FEM) is tested in one-dimensional discontinuous elastic wave propagation. The B-spline based FEM (called Isogeometric analysis IGA) uses spline functions as testing and shape functions in the Galerkin continuous content. Here, the accuracy of stress distribution and spurious oscillations of the B-spline based FEM are studied in numerical modeling of one-dimensional propagation of stress discontinuities in a bar, where the analytical solution is known. For time integration, the Newmark method, implicit form of the generalized-α method, the central difference method and the predictor/multi-corrector method are tested and compared. The use of lumped and consistent mass matrices in explicit time integration is discussed. Due to accuracy, the consistent mass matrix is preferred in explicit time integration in IGA.     

Solution domain decomposition with finite difference methods for partial differential equations

The Solution Domain Decomposition method of Nguyen and Paik [J. Sci. Comput. 4 , 357 (1993)] originally developed in a pseudospectral context is extended for use with finite difference techniques to solve partial differential equations. The essential idea behind this method lies in an application of the superposition principle, which allows interactions between adjacent subdomains to be decoupled and the resulting equations to be solved in parallel. Several tests are performed to assess its accuracy and efficiency based on a model problem arising from thermal convection inside a fluid-saturated porous cavity with heating from a side. The results reveal a well adaptation of the methodology into the framework of finite differences. © 1995 John Wiley & Sons, Inc.     

Refined mixed finite element method for the elasticity problem in a polygonal domain

The purpose of this article is to study a mixed formulation of the elasticity problem in plane polygonal domains and its numerical approximation. In this mixed formulation the strain tensor is introduced as a new unknown and its symmetry is relaxed by a Lagrange multiplier, which is nothing else than the rotation. Because of the corner points, the displacement field is not regular in general in the vicinity of the vertices but belongs to some weighted Sobolev space. Using this information, appropriate refinement rules are imposed on the family of triangulations in order to recapture optimal error estimates. Moreover, uniform error estimates in the Lamé coefficient λ are obtained for λ large. © 2002 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 18: 323–339, 2002; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/num.10009     

An iterative and parallel solver based on domain decomposition for the h-p version of the finite element method

In this paper, we propose a new interative and parallel solver, based on domain decomposition, for the h-p version of the finite element method in two dimensions. It improves our previous work in two aspects: (1) A subdomain may contain several super-elements of the coarse mesh, thus can be of arbitrary shape and size. This makes the solver more efficient and more flexible in computational practice. (2) The p-version components (i.e., the high order side and internal modes) in every element are treated separately, which results in better parallelism.     

Convergence of an adaptive mixed finite element method for convection-diffusion-reaction equations

We prove the convergence of an adaptive mixed finite element method (AMFEM) for (nonsymmetric) convection-diffusion-reaction equations. The convergence result holds for the cases where convection or reaction is not present in convection- or reaction-dominated problems. A novel technique of analysis is developed by using the superconvergence of the scalar displacement variable instead of the quasi-orthogonality for the stress and displacement variables, and without marking the oscillation dependent on discrete solutions and data. We show that AMFEM is a contraction of the error of the stress and displacement variables plus some quantity. Numerical experiments confirm the theoretical results.     

Two-grid stabilized mixed finite element method for fully discrete reaction-diffusion equations

Two-grid mixed finite element method is proposed based on backward Euler schemes for the unsteady reaction-diffusion equations. The scheme combines with the stabilized mixed finite element scheme by using the lowest equal-order pairs for the velocity and pressure. The space two-grid method is also used to reduce the time consuming. The benefits of this approach are to avoid the higher derivative, but to have more favorable stability, and to get the numerical solution of the two unknown variables simultaneously. Stability analysis and error estimates are given in this work. Finally, the theoretical results are verified by the numerical examples.     

Error estimates for the finite element approximation of linear elastic equations in an unbounded domain

In this paper we present error estimates for the finite element approximation of linear elastic equations in an unbounded domain. The finite element approximation is formulated on a bounded computational domain using a nonlocal approximate artificial boundary condition or a local one. In fact there are a family of nonlocal approximate boundary conditions with increasing accuracy (and computational cost) and a family of local ones for a given artificial boundary. Our error estimates show how the errors of the finite element approximations depend on the mesh size, the terms used in the approximate artificial boundary condition, and the location of the artificial boundary. A numerical example for Navier equations outside a circle in the plane is presented. Numerical results demonstrate the performance of our error estimates.

     

A new family of stable elements for nearly incompressible elasticity based on a mixed Petrov-Galerkin finite element formulation

Summary Adding to the classical Hellinger Reissner formulation another residual form of the equilibrium equation, a new Petrov-Galerkin finite element method is derived. It fits within the framework of a mixed finite element method and is proved to be stable for rather general combinations of stress and displacement interpolations, including equal-order discontinuous stress and continuous displacement interpolations which are unstable within the Galerkin approach. Error estimates are presented using the Babuka-Brezzi theory and numerical results confirm these estimates as well as the good accuracy and stability of the method.Dedicated to Professor Ivo Babuka on the occasion of his sixtieth birthdayPrepared for the conference on: The Impact of Mathematical Analysis on the Solution of Engineering Problems. University of Maryland, September 1986.     

A refined mixed finite element method for the boussinesq equations in polygonal domains

This paper deals with the mixed formulation of the Boussinesqequations in two-dimensional polygonal domains and its numericalapproximation. The steady solution has a singular behaviournear the corner points so that we show that it belongs to appropriateweighted Sobolev spaces. Since uniform meshes lead to a slowconvergence rate, we derive appropriate refinement rules onthe meshes near the corner points in order to restore the quasi-optimalrate of convergence. A numerical test is finally presented whichconfirms the theoretical convergence rates. Received 26 November 1999. Accepted 25 August 2000.     

The penalty method for grid matching in mixed finite element methods

In this paper we prove the possibility of the use of the penalty method for grid matching in mixed finite element methods. We consider the Hermann-Johnson scheme for biharmonic equation. The main idea is to construct a perturbed problem with two parameters which play roles of penalties. The perturbed problem is built by the replacement of essential conditions on the interface in the mixed variational statement with natural conditions that contain parameters. The perturbed problem is discretized by the finite element method. We estimate the norm of the difference between a solution of the discrete perturbed problem and a solution of the initial problem; the obtained estimates depend on the step and the penalties. We give recommendations for the choice of penalties depending on the step.     

Analysis of mixed finite element methods on locally refined grids

Summary We consider the mixed finite element method for locally refined triangulations. A local projection operator is defined to satisfy the commutation property that is required in the general theory of mixed methods. Our results can be applied to every known space of arbitrary order over rectangles or triangles. Optimal-order error estimates and superconvergence for the flux along the Gauss lines are established.     

Coupling of viscous and inviscid Stokes equations via a domain decomposition method for finite elements

Summary A generalized Stokes problem is addressed in the framework of a domain decomposition method, in which the physical computational domain is partitioned into two subdomains ₁ and ₂.Three different situations are covered. In the former, the viscous terms are kept in both subdomains. Then we consider the case in which viscosity is dropped out everywhere in . Finally, a hybrid situation in which viscosity is dropped out only in ₁ is addressed. The latter is motivated by physical applications.In all cases, correct transmission conditions across the interface between ₁ and ₂ are devised, and an iterative procedure involving the successive resolution of two subproblems is proposed.The numerical discretization is based upon appropriate finite elements, and stability and convergence analysis is carried out.We also prove that the iteration-by-subdomain algorithms which are associated with the various domain decomposition approaches converge with a rate independent of the finite element mesh size.This work was partially supported by CIRA S.p.A. under the contract Coupling of Euler and Navier-Stokes equations in hypersonic flowsDeceased     

A stabilized Crank-Nicolson mixed finite element method for non-stationary parabolized Navier-Stokes equations

In this study, a time semi-discrete Crank-Nicolson (CN) formulation with second-order time accuracy for the non-stationary parabolized Navier-Stokes equations is firstly established. And then, a fully discrete stabilized CN mixed finite element (SCNMFE) formulation based on two local Gauss integrals and parameterfree with the second-order time accuracy is established directly from the time semi-discrete CN formulation. Thus, it could avoid the discussion for semi-discrete SCNMFE formulation with respect to spatial variables and its theoretical analysis becomes very simple. Finaly, the error estimates of SCNMFE solutions are provided.