34 for the model problem 311 k m will be sparse and

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Unformatted text preview: l. Basis construction is described in Chapter 4, mesh generation and assembly appear in Chapter 5, error analysis is discussed in Chapter 7, and linear algebraic solution strategies are presented in Chapter 11. Problems 1. By introducing the transformation u=u; ^ show that (3.1.1) can be changed to a problem with homogeneous essential boundary conditions. Thus, we can seek u 2 H01. ^ 2. Another method of treating essential boundary conditions is to remove them by using a \penalty function." Penalty methods are rarely used for this purpose, but they are important for other reasons. This problem will introduce the concept and reinforce the material of Section 3.1. Consider the variational statement (3.1.6) as an example, and modify it by including the essential boundary conditions A(v u) = (v f )+ < v >@ N + < v ; u >@ E 8v 2 H 1 : Here is a penalty parameter and subscripts on the boundary integral indicate their domain. No boundary conditions are applied and the problem is solved for u and v ranging over the whole of H 1. 3.3. Overview of the Finite Element Method 21 Show that smooth solutions of this variational problem satisfy the di erential equation (3.1.1a) as well as the natural boundary conditions (3.1.1c) and @u u + p @n = (x y) 2 E: The penalty parameter must be selected large enough for this natural boundary condition to approximate the prescribed essential condition (3.1.1b). This can be tricky. If selected too large, it will introduce ill-conditioning into the resulting algebraic system. 22 Multi-Dimensional Variational Principles Bibliography 1] R.A. Adams. Sobolev Spaces. Academic Press, New York, 1975. 2] O. Axelsson and V.A. Barker. Finite Element Solution of Boundary Value Problems. Academic Press, Orlando, 1984. 3] C. Geo man and G. Pedrick. First Course in Functional Analysis. Prentice-Hall, Englewood Cli s, 1965. 4] P.R. Halmos. Measure Theory. Springer-Verlag, New York, 1991. 5] J.T. Oden and L.F. Demkowicz. Applied Functional Analysis. CRC Press, Boca Raton, 1996. 6] R. Wait and A.R. Mitchell. The Finite Element Analysis and Applications. John Wiley and Sons, Chichester, 1985. 23...
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This document was uploaded on 03/16/2014 for the course CSCI 6860 at Rensselaer Polytechnic Institute.

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