Über ein Variationsprinzip zur Lösung von Dirichlet-Problemen bei Verwendung von Teilräumen, die keinen Randbedingungen unterworfen sind

This paper concerns with numerical approximations of solutions of fully nonlinear second order partial differential equations (PDEs). A new notion of weak solutions, called moment solutions, is introduced for fully nonlinear second order PDEs. Unlike viscosity solutions, moment solutions are defined by a constructive method, called the vanishing moment method, and hence, they can be readily computed by existing numerical methods such as finite difference, finite element, spectral Galerkin, and discontinuous Galerkin methods. The main idea of the proposed vanishing moment method is to approximate a fully nonlinear second order PDE by a higher order, in particular, a quasilinear fourth order PDE. We show by various numerical experiments the viability of the proposed vanishing moment method. All our numerical experiments show the convergence of the vanishing moment method, and they also show that moment solutions coincide with viscosity solutions whenever the latter exist.

https://www.cct.lsu.edu/~neilan/Neilan/Research_files/VMMfulltext.pdf

Vanishing Moment Method and Moment Solutions for Fully Nonlinear Second Order Partial Differential Equations

Dynamic large deflection analysis of plates using mixed finite elements

Publisher Summary This chapter focuses on Dirichlet problems using subspaces with nearly zero boundary conditions. The classical Rayleigh-Ritz methods for elliptic boundary value problems with Dirichlet data assume that the approximating functions fulfill the boundary conditions exactly. In most cases, that is, when the boundary is curved, the construction of such subspaces is delicate and implies many difficulties in practice. To overcome this problem there are now known modifications of the variational principles obtained by adding appropriate boundary terms. In these cases the approximating subspaces which are used need not fulfill any boundary conditions. The chapter presents versions of the Ritz and least squares methods and shows them to be quasi-optimal. Besides the concept of nearly zero boundary values and a certain inverse assumption, a sharper condition of approximability than normally used is needed. The cubic spline functions are also discussed in the chapter.

On dirichlet problems using subspaces with nearly zero boundary conditions

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

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

A tinite element method of mixed type is proposed to solve the Dirichlet problem of the von Karman equations. Existence and convergence of the approximate solution are proved.

A mixed finite element method for the solution of the von Kármán equations

Foundations of the numerical analysis of plasticity

where D is a bounded domain in the (#, jO -plane, P is the boundary of D, and 0), &(:>0), c(>0), / are smooth functions defined on D. In Ritz-Galerkin's method, first we choose a system of linearly independent functions {^J such that they satisfy the given homogeneous boundary condition and they are dense in a function space containing the exact solution of the above boundary value problem, and next we seek the m-th approximation um in the form

https://www.ems-ph.org/journals/show_pdf.php?issn=0034-5318&vol=4&iss=1&rank=6

On the Convergence of Ritz-Galerkin's Method

https://www.ems-ph.org/journals/show_pdf.php?issn=0034-5318&vol=16&iss=2&rank=4

Elastic-Plastic Vibration of a Rod

Publisher Summary This chapter discusses yielding and unloading in semidiserete problem of plasticity. The transition from the elastic state to the plastic state is called “yielding.” The reverse change is a case of unloading. Therefore, the material undergoes the elastic and plastic deformations repeating yielding and unloading alternately. For the semidiscrete system––that is, for the case that only the spatial region is discrete––it is possible to show the existence of the classical solution. The chapter presents the way to get this solution.This kind of systems is essential for deriving the approximate methods in engineering. Especially, the explicit integration schemes start from this system. Also, if the solution of the fully continuous problem is needed, it is only necessary to pass to the limit with respect to the spatial approximation. The key to prove the existence of the classical solution is to guess the behavior of the solution after the time at which the state of the element may change.

Tetsuhiko Miyoshi

Papers

A mixed finite element method for the solution of the von Kármán equations

Foundations of the numerical analysis of plasticity

On the Convergence of Ritz-Galerkin's Method

Elastic-Plastic Vibration of a Rod

Yielding and Unloading in Semidiscrete Problem of Plasticity