Über ein Variationsprinzip zur Lösung von Dirichlet-Problemen bei Verwendung von Teilräumen, die keinen Randbedingungen unterworfen sind
01 Jul 1971-Abhandlungen Aus Dem Mathematischen Seminar Der Universitat Hamburg (Springer-Verlag)-Vol. 36, Iss: 1, pp 9-15
About: This article is published in Abhandlungen Aus Dem Mathematischen Seminar Der Universitat Hamburg.The article was published on 1971-07-01. It has received 1450 citations till now. The article focuses on the topics: Differential geometry.
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TL;DR: In this paper, a framework for the analysis of a large class of discontinuous Galerkin methods for second-order elliptic problems is provided, which allows for the understanding and comparison of most of the discontinuous methods that have been proposed over the past three decades.
Abstract: We provide a framework for the analysis of a large class of discontinuous methods for second-order elliptic problems. It allows for the understanding and comparison of most of the discontinuous Galerkin methods that have been proposed over the past three decades for the numerical treatment of elliptic problems.
3,293 citations
15 Nov 2004
TL;DR: The mathematical structure of the contact formulation for finite element methods is derived on the basis of a continuum description of contact, and several algorithms related to spatial contact search and fulfillment of the inequality constraints at the contact interface are discussed.
Abstract: This paper describes modern techniques used to solve contact problems within Computational Mechanics. On the basis of a continuum description of contact, the mathematical structure of the contact formulation for finite element methods is derived. Emphasis is also placed on the constitutive behavior at the contact interface for normal and tangential (frictional) contact. Furthermore, different discretization schemes currently applied to solve engineering problems are formulated for small and finite strain problems. These include isoparametric interpolations, node-to-segment discretizations and also mortar and Nitsche techniques. Furthermore, several algorithms related to spatial contact search and fulfillment of the inequality constraints at the contact interface are discussed. Here, especially the penalty and Lagrange multiplier schemes are considered and also SQP- and linear-programming methods are reviewed.
Keywords:
contact mechanics;
friction;
penalty method;
Lagrange multiplier method;
contact algorithms;
finite element method;
finite deformations;
discretization methods
1,761 citations
TL;DR: In this paper, a semidiscrete finite element method for the solution of second order nonlinear parabolic boundary value problems is formulated and analyzed, where the test and trial spaces consist of discontinuous piecewise polynomial functions over quite general meshes with interelement continuity enforced approximately by means of penalties.
Abstract: A new semidiscrete finite element method for the solution of second order nonlinear parabolic boundary value problems is formulated and analyzed. The test and trial spaces consist of discontinuous piecewise polynomial functions over quite general meshes with interelement continuity enforced approximately by means of penalties. Optimal order error estimates in energy and $L^2$-norms are stated in terms of locally expressed quantities. They are proved first for a model problem and then in general.
1,607 citations
TL;DR: In this article, the Dirichlet problem for second order differential equations is chosen as a model problem to show how the finite element method may be implemented to avoid difficulty in fulfilling essential (stable) boundary conditions.
Abstract: The Dirichlet problem for second order differential equations is chosen as a model problem to show how the finite element method may be implemented to avoid difficulty in fulfilling essential (stable) boundary conditions. The implementation is based on the application of Lagrangian multiplier. The rate of convergence is proved.
1,579 citations
TL;DR: In this article, two families of mixed finite elements, one based on triangles and the other on rectangles, are introduced as alternatives to the usual Raviart-Thomas-Nedelec spaces.
Abstract: Two families of mixed finite elements, one based on triangles and the other on rectangles, are introduced as alternatives to the usual Raviart-Thomas-Nedelec spaces. Error estimates inL 2 (Ω) andH ?5 (Ω) are derived for these elements. A hybrid version of the mixed method is also considered, and some superconvergence phenomena are discussed.
1,213 citations
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Book•
01 Jan 1965TL;DR: In this article, the authors present an introduction to the theory of higher-order elliptic boundary value problems, and a detailed study of basic problems of the theory, such as the problem of existence and regularity of solutions of higher order elliptic edge value problems.
Abstract: This book, which is a new edition of a book originally published in 1965, presents an introduction to the theory of higher-order elliptic boundary value problems. The book contains a detailed study of basic problems of the theory, such as the problem of existence and regularity of solutions of higher-order elliptic boundary value problems. It also contains a study of spectral properties of operators associated with elliptic boundary value problems. Weyl's law on the asymptotic distribution of eigenvalues is studied in great generality.
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