Techn. Universiẗat, Diss., 2003--Clausthal.

University, Diss., 2004--Bonn.

Thesis (M.S.)--Brigham Young University. Dept of Mathematics, 2006. / Includes bibliographical references (p. 94-96).

The primary goal of this thesis is to study finite element based a priori and a posteriori error estimates of optimal control problems of various kinds governed by linear elliptic PDEs (partial differential equations) of second and fourth orders. This thesis studies interior and boundary control (Neumann and Dirichlet) problems. The initial chapter is introductory in nature. Some preliminary and fundamental results of finite element methods and optimal control problems which play key roles for the subsequent analysis are reviewed in this chapter. This is followed by a brief literature survey of the finite element based numerical analysis of PDE constrained optimal control problems. We conclude the chapter with a discussion on the outline of the thesis. An abstract framework for the error analysis of discontinuous Galerkin methods for control constrained optimal control problems is developed in the second chapter. The analysis establishes the best approximation result from a priori analysis point of view and delivers a reliable and efficient a posteriori error estimator. The results are applicable to a variety of problems just under the minimal regularity possessed by the well-posedness of the problem. Subsequently, the applications of p p - interior penalty methods for a boundary control problem as well as a distributed control problem governed by the bi-harmonic equation subject to simply supported boundary conditions are discussed through the abstract analysis. In the third chapter, an alternative energy space based approach is proposed for the Dirichlet boundary control problem and then a finite element based numerical method is designed and analyzed for its numerical approximation. A priori error estimates of optimal order in the energy norm and the m norm are derived. Moreover, a reliable and efficient a posteriori error estimator is derived with the help an auxiliary problem. An energy space based Dirichlet boundary control problem governed by bi-harmonic equation is investigated and subsequently a l y - interior penalty method is proposed and analyzed for it in the fourth chapter. An optimal order a priori error estimate is derived under the minimal regularity conditions. The abstract error estimate guarantees optimal order of convergence whenever the solution has minimum regularity. Further an optimal order l l norm error estimate is derived. The fifth chapter studies a super convergence result for the optimal control of an interior control problem with Dirichlet cost functional and governed by second order linear elliptic PDE. An optimal order a priori error estimate is derived and subsequently a super convergence result for the optimal control is derived. A residual based reliable and efficient error estimators are derived in a posteriori error control for the optimal control. Numerical experiments illustrate the theoretical results at the end of every chapter. We conclude the thesis stating the possible extensions which can be made of the results presented in the thesis with some more problems of future interest in this direction.

Estudamos o problema de Dirichlet para a equação das superfícies mínimas em domínios limitados do plano. Provamos a existência e unicidade de gráficos mínimos sobre domínios limitados e não necessariamente convexos, com valores no bordo satisfazendo uma condição que denominamos condição da declividade limitada generalizada a qual, usando cilindros no lugar de planos, generaliza a condição clássica da declividade limitada. Com este resultado, dado um domínio limitado e suave qualquer do plano, conseguimos obter cotas explícitas para a norma C2 de dados no bordo deste domínio que garantem a existência de solução ao correspondente problema de Dirichlet.

SILVA, José Deibsom da; BESSA, Gregório Pacelli Feitosa. Uma extensão do teorema de Barta e aplicações geométricas. 2010. 41 f. :Dissertação (mestrado)- Universidade Federal do Ceará, Pós-Graduação em Matemática, Fortaleza-CE, 2010. / Submitted by Rocilda Sales (rocilda@ufc.br) on 2011-10-27T13:03:47Z No. of bitstreams: 1 2010_dis_jdsilva.pdf: 275124 bytes, checksum: 8ea6b5ce69a064049ca7d6d9ee9edcdd (MD5) / Approved for entry into archive by Rocilda Sales(rocilda@ufc.br) on 2011-10-27T13:05:38Z (GMT) No. of bitstreams: 1 2010_dis_jdsilva.pdf: 275124 bytes, checksum: 8ea6b5ce69a064049ca7d6d9ee9edcdd (MD5) / Made available in DSpace on 2011-10-27T13:05:38Z (GMT). No. of bitstreams: 1 2010_dis_jdsilva.pdf: 275124 bytes, checksum: 8ea6b5ce69a064049ca7d6d9ee9edcdd (MD5) Previous issue date: 2010 / We present an extension to Barta's Theorem due to G. P. Bessa and J. F. Montenegro and we show some geometric applications of the obtained result. As first application, we extend Chang's lower eigenvalue estimates of the Laplacian in normal geodesic balls. As second application, we generalize Cheng-Li-Yau's eigenvalue estimates to a minimal submanifold of the space forms. / Apresentamos uma extensão do Teorema de Barta devido a G. P. Bessa and J. F. Montenegro e fazemos algumas aplicações geométricas do resultado obtido. A primeira aplicação geométrica da extensão do Teorema de Barta é uma extensão do Teorema de Cheng sobre estimativas inferiores de autovalores do Laplaciano em bolas geodésicas normais. A segunda aplicação geométrica é uma generalização do Teorema de Cheng-Li-Yau de estimativas de autovalores para uma subvariedade mínima do espaço forma.

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Estudamos o problema de Dirichlet para a equação das superfícies mínimas em domínios limitados do plano. Provamos a existência e unicidade de gráficos mínimos sobre domínios limitados e não necessariamente convexos, com valores no bordo satisfazendo uma condição que denominamos condição da declividade limitada generalizada a qual, usando cilindros no lugar de planos, generaliza a condição clássica da declividade limitada. Com este resultado, dado um domínio limitado e suave qualquer do plano, conseguimos obter cotas explícitas para a norma C2 de dados no bordo deste domínio que garantem a existência de solução ao correspondente problema de Dirichlet.

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