A Least-Squares Mixed Finite Element for Quasi-Incompressible Elastodynamics

In the present work, a mixed finite element based on a modified least-squares formulation is proposed. Here, we consider the time-dependent equations for quasi-incompressible elastodynamics under small strain assumptions. The main goal is to obtain an accurate approximation of both displacements and stresses in particular for the lowest-order element. Basis for the element formulation is a weak form resulting from a least-squares method. The L₂-norm minimization of the time-discretized residuals of the given first-order system leads to a functional depending on approximations for displacements and stresses. By introducing a time-independent displacement test function, a weak form is derived. A numerical example concerning quasi-incompressible elasticity shows the performance of the approach for the lowest-order element RT₀P₁as. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A Weighted Least-Squares Mixed Finite Element Formulation for Quasi-Incompressible Elastodynamics

The purpose of this work is the application of the least-squares finite element method to an elastodynamic, quasi-incompressible problem under small strain assumptions. Therefore a mixed finite element based on a weighted least-squares formulation is developed. The L₂-norm minimization of the time-discretized residuals of the given first-order system of partial differential equations leads to a functional depending on displacements and stresses. In the numerical example the proposed mixed element is compared to an alternative approach, which is based on a least-squares mixed finite element with improved momentum balance, see [1]. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Remarks on a Least Squares Mixed Finite Element for Plane Elasticity

The objective of this work is to discuss a least squares finite element method within plane elasticity problems. The L ₂-norm minimization of the residuals of the given first order system of differential equations leads to a functional, which is a two field formulation in the displacements and the stresses. The governing equations for the considered least squares mixed finite element are derived. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Comparison of geometrically nonlinear LSFEM formulations based on different hyperelastic models

This contribution deals with the solution of geometrically nonlinear elastic problems solved by the least-squares mixed finite element method (LSFEM). The degrees of freedom (displacements and stresses) will be approximated using suitable spaces, namely W^1,p with p > 4 and H(div,Ω). In order to define the stress response of the material, different hyperelastic free energy functions will be presented. The residual forms ℛ_I of the balance of momentum and the constitutive equation build a system of differential equations of first order. Choosing suitable weighting operators and applying L₂-norms lead to a least-squares functional ℱ( P,u ). The interpolation of the unknowns is accomplished using a standard polynomial interpolation for the displacements and vector-valued Raviart-Thomas functions for the approximation of the stresses. The formulations presented will be compared considering a uni-axial spatial tension test. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

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.     

Independence principles in the equations of state of a deformable material

We consider the principles of coordinate, rotational, and initial independence of the equations of state for a deformable material and the theorem on the existence of elasticity potential connected with them. We show that the well-known axiomatic substantiation and mathematical representation of these principles in “rational continuum mechanics” as well as the proof of the theorem are erroneous. A correct proof of the principles and theorem is presented for the most general case (a stressed anisotropic body under the action of an arbitrary tensor field) without applying any axioms. On this basis, we eliminated the dependence on an arbitrary initial state and the corresponding accumulated strain from the system of equations of state of a deformable material. The obtained forms of equations are convenient for constructing and analyzing the equations of local influence of initial stresses on physical fields of different nature. Finally, these equations represent governing equations for the problems of nondestructive testing of inhomogeneous three-dimensional stress fields and for theoretical-and-experimental investigation of the nonlinear equations of state.     

Solving hyperelastic problems using mixed LSFEM

The focus of this contribution is the solution of hyperelastic problems using the least-squares finite element method (LSFEM). In particular a mixed least-squares finite element formulation is provided and applied on geometrically nonlinear problems. The basis for the element formulation is a div-grad first-order system consisting of the equilibrium condition and the constitutive equation both written in a residual form. An L₂-norm is adopted on the residuals leading to a functional depending on displacements and stresses which has to be minimized. Therefore the first variations with respect to both free variables have to be zero. The solution can then be found by applying Newton's Method. For the continuous approximation of the displacements in W^1,p with p > 2, standard polynomials are used. Shape functions belonging to a Raviart-Thomas space are applied for the stress interpolation. These vector-valued functions ensure a conforming discretization of the Sobolev space H(div, Ω). Finally the proposed formulation is tested in a numerical example. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A two-stage least-squares finite element method for the stress-pressure-displacement elasticity equations

A new stress-pressure-displacement formulation for the planar elasticity equations is proposed by introducing the auxiliary variables, stresses, and pressure. The resulting first-order system involves a nonnegative parameter that measures the material compressibility for the elastic body. A two-stage least-squares finite element procedure is introduced for approximating the solution to this system with appropriate boundary conditions. It is shown that the two-stage least-squares scheme is stable and, with respect to the order of approximation for smooth exact solutions, the rates of convergence of the approximations for all the unknowns are optimal both in the H¹-norm and in the L²-norm. Numerical experiments with various values of the parameter are examined, which demonstrate the theoretical estimates. Among other things, computational results indicate that the behavior of convergence is uniform in the nonnegative parameter. © 1998 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 14: 297–315, 1998     

Extinction of solutions of parabolic equations with variable anisotropic nonlinearities

We study the Dirichlet problem for a class of nonlinear parabolic equations with nonstandard anisotropic growth conditions that generalize the evolutional p(x, t)-Laplacian. We study the property of extinction of solutions in finite time. In particular, we show that the extinction may take place even in the borderline case when the equation becomes linear as t → ∞.     

Error analysis of a weighted least-squares finite element method for 2-D incompressible flows in velocity–stress–pressure formulation

In this paper we are concerned with a weighted least-squares finite element method for approximating the solution of boundary value problems for 2-D viscous incompressible flows. We consider the generalized Stokes equations with velocity boundary conditions. Introducing the auxiliary variables (stresses) of the velocity gradients and combining the divergence free condition with some compatibility conditions, we can recast the original second-order problem as a Petrovski-type first-order elliptic system (called velocity–stress–pressure formulation) in six equations and six unknowns together with Riemann–Hilbert-type boundary conditions. A weighted least-squares finite element method is proposed for solving this extended first-order problem. The finite element approximations are defined to be the minimizers of a weighted least-squares functional over the finite element subspaces of the H¹ product space. With many advantageous features, the analysis also shows that, under suitable assumptions, the method achieves optimal order of convergence both in the L²-norm and in the H¹-norm. © 1998 B. G. Teubner Stuttgart—John Wiley & Sons, Ltd.     

An L2 finite element approximation for the incompressible Navier–Stokes equations

This paper utilizes the Picard method and Newton's method to linearize the stationary incompressible Navier–Stokes equations and then uses an LL* approach, which is a least-squares finite element method applied to the dual problem of the corresponding linear system. The LL* approach provides an L²-approximation to a given problem, which is not typically available with conventional finite element methods for nonlinear second-order partial differential equations. We first show that the proposed combination of linearization scheme and LL* approach provides an L²-approximation to the stationary incompressible Navier–Stokes equations. The validity of L²-approximation is proven through the analysis of the weak problem corresponding to the linearized Navier–Stokes equations. Then, the convergence is analyzed, and numerical results are presented.     

Two-Grid hp-Version DGFEMs for Strongly Monotone Second-Order Quasilinear Elliptic PDEs

In this article we develop the a priori error analysis of so-called two-grid hp-version discontinuous Galerkin finite element methods for the numerical approximation of strongly monotone second-order quasilinear partial differential equations. In this setting, the fully nonlinear problem is first approximated on a coarse finite element space V(𝒯_H, P ). The resulting ‘coarse’ numerical solution is then exploited to provide the necessary data needed to linearize the underlying discretization on the finer space V(𝒯_h, p ); thereby, only a linear system of equations is solved on the richer space V(𝒯_h, p ). Numerical experiments confirming the theoretical results are presented. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

The presentation of explicit analytical solutions of a class of nonlinear evolution equations

In this paper, we introduce a function set Ω_m. There is a conjecture that an arbitrary explicit travelling-wave analytical solution of a real constant coefficient nonlinear evolution equation is necessarily a linear (or nonlinear) combination of the product of some elements in Ω_m. A widespread applicable approach for solving a class of nonlinear evolution equations is established. The new analytical solutions to two kinds of nonlinear evolution equations are described with the aid of the guess.     

Global well‐posedness of 2D nonlinear Boussinesq equations with mixed partial viscosity and thermal diffusivity

In this paper, we discuss with the global well‐posedness of 2D anisotropic nonlinear Boussinesq equations with any two positive viscosities and one positive thermal diffusivity. More precisely, for three kinds of viscous combinations, we obtain the global well‐posedness without any assumption on the solution. For other three difficult cases, under the minimal regularity assumption, we also derive the unique global solution. To the authors' knowledge, our result is new even for the simplified model, that is, F(θ) = θe₂. Copyright © 2017 John Wiley & Sons, Ltd.     

Free Boundary Value Problems for Abstract Elliptic Equations and Applications

The free boundary value problems for elliptic differential-operator equations are studied. Several conditions for the uniform maximal regularity with respect to boundary parameters and the Fredholmness in abstract L _p -spaces are given. In application, the nonlocal free boundary problems for finite or infinite systems of elliptic and anisotropic type equations are studied.     

Dissipative Dynamics for a Class of Nonlinear Pseudo-Differential Equations

We study a class of nonlinear evolutionary equations generated by an elliptic pseudo-differential operator, and with nonlinearity of the form G(u _x ) where cη² ≤ G(η) ≤ Cη² for large |η|. For the evolution in spaces of periodic functions with zero mean we demonstrate existence of a universal absorbing set and compact attractor. Furthermore, we show that the attractor is of a finite Hausdorf dimension. The dissipation mechanism for the class of equations studied in the paper is akin to the nonlinear saturation in the Kuramoto-Sivashinsky equation. A similar generalization of the Kuramoto-Sivashinsky equation was studied by Nicolaenko et al. under the assumption of a purely quadratic nonlinearity and reflection invariance of both: the equation and solutions.      

Positive solutions for nonlinear operator equations and several classes of applications

In this paper, we study a class of nonlinear operator equations x = Ax + x ₀ on ordered Banach spaces, where A is a monotone generalized concave operator. Using the properties of cones and monotone iterative technique, we establish the existence and uniqueness of solutions for such equations. In particular, we do not demand the existence of upper-lower solutions and compactness and continuity conditions. As applications, we study first-order initial value problems and two-point boundary value problems with the nonlinear term is required to be monotone in its second argument. In the end, applications to nonlinear systems of equations and to nonlinear matrix equations are also considered.     

Investigation of stress-velocity LSFEMs for the incompressible Navier-Stokes equations

In this contribution three mixed least-squares finite element methods (LSFEMs) for the incompressible Navier-Stokes equations are investigated with respect to accuracy and efficiency. The well-known stress-velocity-pressure formulation is the basis for two further div-grad least-squares formulations in terms of stresses and velocities (SV). Advantage of the SV formulations is a system with a smaller matrix size due to a reduction of the degrees of freedom. The least-squares finite element formulations, which are investigated in this contribution, base on the incompressible stationary Navier-Stokes equations. The first formulation under consideration is the stress-velocity-pressure formulation according to [1]. Secondly, an extended stress-velocity formulation with an additional residual is derived based on the findings in [1] and [5]. The third formulation is a pressure reduced stress-velocity formulation based on a condensation scheme. Therefore, the pressure is interpolated discontinuously, and eliminated on the discrete level without the need for any matrix inverting. The modified lid-driven cavity boundary value problem, is investigated for the Reynolds number Re = 1000 for all three formulations. (© 2017 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Operator-valued Chandrasekhar H-functions

Operator valued analogs of the Chandrasekhar H-function, that occur in the study of neutron transport in a slab with continuous energy dependence and anisotropic scattering, satisfy a system of nonlinear integral equations. An appropriate Banach space setting is found for the study of this system. We show that the system may be solved by iteration. We extend the domain of analyticity of H_r and H_t by means of bifurcation theory.     

An adaptive least-squares mixed finite element method for nonlinear parabolic problems

A least-squares mixed finite element method for nonlinear parabolic problems is investigated in terms of computational efficiency. An a posteriori error estimator, which is needed in an adaptive refinement algorithm, was composed with the least-squares functional, and a posteriori errors were effectively estimated.