A complete two-phase numerical model of film condensation from a mixture of a vapour and a non-condensing gas that is based on the two-dimensional elliptic governing equations with variable physical properties is presented. The model predicts the full viscous flow and heat and mass transfer for the mixture around the tube and in the entire liquid film from the top of the tube to the falling film below the tube. A finite volume method is used with a segregated solution approach and a dynamically moving computational grid that tracks the phase interface sharply. Fundamental balances of mass, energy, and force are enforced accurately at the phase interface. The model was developed in steps and validated against various experimental and theoretical works in the literature for different two-phase flows. The validation tests included stratified flow of liquid and gas in a horizontal channel, falling liquid film over a vertical wall, and condensation of steam from a steam-air mixture in a vertical channel. The model was used to simulate laminar film condensation from a downward flowing steam-air mixture over an isothermal horizontal tube. The validity of this new model is demonstrated by comparisons with previous theoretical and experimental studies. New results are presented on the effects of free-stream-to-tube temperature difference, upstream Reynolds number, free-stream gas mass fraction, and free-stream pressure on the condensate film development, the local and average heat transfer coefficients, and the total condensate mass flow rate. It was found that the temperature difference had the greatest effect on the condensation rate and film thickness. The presence of non-condensing gas in the vapour has a strong negative impact on the condensation process. For the pure steam case, moderate changes in the upstream Reynolds number showed slight increases in condensate mass flow rate with increased Reynolds number. For the mixture case, however, moderate increase in upstream Reynolds number increases significantly the condensate mass flow rate and film thickness. This trend becomes more noticeable as the free-steam gas mass fraction increases. Changing the free-stream pressure demonstrated that property variation had a relatively smaller effect than temperature difference and gas mass fraction changes. / February 2017

Este trabalho propõe um esquema de diferenças finitas compacta de alta ordem para resolver problemas elípticos com coeficientes variáveis em malhas composta. São apresentados a formulação matemática e a dedução do método compacto de quarta ordem aplicado à problemas elípticos bidimensionais, em malha regular e composta. Foi adotado o uso da biblioteca PETSc com os seus pré-condicionadores e métodos numéricos para resolver os sistemas lineares resultantes da discretização do problema. Por fim, testes visando verificar o código foram feitos, utilizando o método de soluções manufaturadas, para mostrar alta eficiência e acurácia do método desenvolvido. / This paper proposes a scheme of compact finite difference higher order for solve elliptic problems with variable coeficients in composite meshes. we present the mathematical formulation and the deduction of the compact method of fourth order applied to two-dimensional elliptic problems in regular and composite mesh . It was adopted using the PETSc library with its pre- conditioners and numerical methods for solving linear systems resulting from discretization of the problem. Finally , tests to verify the code were made using the method of manufactured solutions to show high eficiency and accuracy of the method developed .

In the dissertation we discuss the rectangular elements, Adini's elements and $p-$order Lagrange elements, which were constructed in the rectangular finite spaces. The special rectangular partitions enable the finite element solutions $u_h$ more efficient in interpolation of the true solution for Elliptic equation $u_I$. The convergence rates of $|u_h-u_I|_1$ are one or two orders higher than the optimal convergence rates. For post-processings we construct higher order interpolation operation $Pi_p$ to reach superconvergence $|u-Pi_p u_h|_1$. To our best knowledge, we at the first time provided the a posteriori interpolant formulas of Adini's elements and biquadratic Lagrange elements to obtain the global superconvergence, and at the first time reported the numerical verification for supercloseness $O(h^4)-O(h^5) $, global superconvergence $O(h^5)$ in $H^1$-norm and the high rates $O(h^6|ln h|)$ in the infinity norm for Poisson's equation(i.e., $-Delta u = f$). Since the finite element methods is fail to deal with the singularity problems, in the dissertation, the combinations of the Ritz-Galerkin method and the finite element methods are used for the singularity problem, i.e., Motz's problem. To couple two methods along their common boundary, we adopt the simplified hybrid, penalty, and penalty plus hybrid techniques. The analysis are made in the dissertation to derive the almost best global superconvergence $O(h^{p+2-delta})$ in $H^1$-norm, $0<delta << 1$, for the combination using $p(geq 2)$-rectangles in the smooth subdomain, and the best global superconvergence $O(h^{3.5})$ in $H^1$-norm for combinations of Adini's elements in the smooth subdomain. The numerical experiments have been carried out for the combinations of the Ritz-Galerkin method and Adini's elements, to verify the theoretical superconvergence derived.

For degenerate elliptic partial differential equations, it is often desirable to show that a weak solution is smooth. The first and most difficult step in this process is establishing local Hölder continuity. Sufficient conditions for establishing continuity have already been documented in [FP], [SW1], and [MRW], and their necessity in [R]. However, the complexity of the equations discussed in those works makes it difficult to understand the core structure of the arguments employed. Here, we present a harmonic-analytic method for establishing Hölder continuity of weak solutions in context of a simple linear equation div(Q?u) = f in a homogeneous space structure in order to showcase the form of the argument. Ad- ditionally, we correct an oversight in the adaptation of the John-Nirenberg inequality presented in [SW1], restricting it to a much smaller class of balls.

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Submitted by Johnny Rodrigues (johnnyrodrigues@ufcg.edu.br) on 2018-07-05T18:49:14Z No. of bitstreams: 1 JOSÉ FERNANDO LEITE AIRES - DISSERTAÇÃO PPGMAT 2004..pdf: 619280 bytes, checksum: fd21b35d13e1bed399affca7c1d08370 (MD5) / Made available in DSpace on 2018-07-05T18:49:14Z (GMT). No. of bitstreams: 1 JOSÉ FERNANDO LEITE AIRES - DISSERTAÇÃO PPGMAT 2004..pdf: 619280 bytes, checksum: fd21b35d13e1bed399affca7c1d08370 (MD5) Previous issue date: 2004-03 / Capes / Para visualização completa do resumo recomendamos o download do arquivo, uma vez que o mesmo possui fórmulas de equações que não foram possíveis copia-las aqui. / For a complete preview of the summary we recommend downloading the file, since it has formulas of equations that could not be copied here.

Submitted by Johnny Rodrigues (johnnyrodrigues@ufcg.edu.br) on 2018-07-25T17:05:28Z No. of bitstreams: 1 DENILSON DA SILVA PEREIRA - DISSERTAÇÃO PPGMAT 2010..pdf: 630527 bytes, checksum: 8a6ec5b5fb5e2a462945183d2180a573 (MD5) / Made available in DSpace on 2018-07-25T17:05:28Z (GMT). No. of bitstreams: 1 DENILSON DA SILVA PEREIRA - DISSERTAÇÃO PPGMAT 2010..pdf: 630527 bytes, checksum: 8a6ec5b5fb5e2a462945183d2180a573 (MD5) Previous issue date: 2010-12 / Neste trabalho estudamos a existência e multiplicidade de soluções para uma certa classe de problemas elípticos. Utilizaremos métodos variacionais juntamente com a teoria de Morse em dimensão infinita. / In this work, we study the existence and multiplicity of solution for a large class of Elliptic problems. The main tools used are variational methods together with the infinite dimensional Morse Theory.

Made available in DSpace on 2015-03-26T13:45:33Z (GMT). No. of bitstreams: 1 texto completo.pdf: 346839 bytes, checksum: cab5395001fcc113256f79ba4e365ce8 (MD5) Previous issue date: 2010-02-26 / In this work we study two class of elliptic problems modeled on unbounded domains. The study of these class of problems is relevant not only in applied mathematics, but also in nonlinear analysis. In the these problems, since the domain is unbounded, there is a lack of compactness of the Sobolev embedding, bringing some difficults to show the convergence of the Palais-Smale sequence. To solve this difficulty we work in a subspace of the usual Sobolev space where we can recover some compactness result. The solutions are obtained by Lagrange multiplier. We give another proof of results in [6] due to Wei-Yue Ding and Wei-Ming Ni, who used to solve The Mountain Pass Theorem and a priori estimates. The results of our study are due to Habao Su, Zhi-Qiang Wang and Michel Willem. / Neste trabalho, estudamos duas classes de problemas elípticos modeladas em domínios ilimitados. O estudo dessas classes de problemas e relevante não só no campo da matemática aplicada, mas também na área de análise não linear. Nesses problemas, como o domínio é ilimitado, há a perda de compacidade da &#8220;imersão" de Sobolev, dificultando a convergência da sequência de &#8220;soluções" (sequência de Palais Smale). Essa dificuldade é contornada trabalhando num subespaço do espaço de Sobolev usual onde se recupera a compacidade utilizando resultados de imersão. As soluções são obtidas via multiplicadores de Lagrange. Apresentamos uma outra maneira de resolver um problema em [6], devido a Wei-Yue Ding e Wei-Ming Ni, que utilizaram na solução o Teorema do Passo da Montanha e estimativas a priori. Os resultados de nosso estudo são devidos a Habao Su, Zhi-Qiang Wang e Michel Willem.

Made available in DSpace on 2015-05-15T11:46:13Z (GMT). No. of bitstreams: 1 arquivototal.pdf: 572294 bytes, checksum: 1c46e916c7cc2e4689880e2687dbee0b (MD5) Previous issue date: 2012-03-08 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / This word is concerned with the study of bounded solutions of semilinear elliptic equations u &#8722; F0(u) = 0 in the whole space Rn, under the assumption that u is monotone in one direction, say, @u/@xn > 0 em Rn. The goal is to establish the one-dimensional character or symmetry of u, namely, that u only depends on one variable or, equivalently, that the level sets of u are hyperplanos. This type of symmetry question was raised by de Giorgi in 1978 (see [6]), who made the folowing conjecture: Conjecture Suppose that u 2 C2(Rn) is solution of the equation u + u &#8722; u3 = 0 satisfying |u(x)| 1 and @u @xn > 0 in the whole Rn. Then the level sets of u must be hyperplanes. We show a stronger version of De Giorgi s conjecture is indeed true in dimension 2 and 3 using some techniques in the linear theory developed by Berestychi, Caffarelli and Nirenberg [5] in one of their papers on qualitative properties of solutions of semilinear elliptic equations. / Este trabalho se preocupa com o estudo de soluções limitadas de equações elípticas semilineares u &#8722; F0(u) = 0 em todo espaço Rn, sob o pressuposto que u é monótona em uma direção, digamos @u/@xn > 0 em Rn. O objetivo é estabelecer o caráter unidimensional ou simetria de u, ou seja, que u depende apenas de uma variável ou equivalentemente, que os conjuntos de nível de u são hiperplanos. Este tipo de questão da simetria foi levantada por De Giorgi em 1978 (ver [6]), que fez a seguinte conjectura: Conjectura Suponha que u 2 C2(Rn) é solução da equação u + u &#8722; u3 = 0 satisfazendo |u(x)| 1 e @u @xn > 0 em todo Rn. Então os conjuntos de nível de u são hiperplanos. Mostraremos que uma versão forte da conjectura de De Giorgi é de fato verdade em dimensão 2 e 3 usando somente técnicas da teoria linear desenvolvida por Berestychi, Caffarelli e Nirenberg [5] em um dos seus artigos sobre as propriedades qualitativas de equações elípticas semilineares.

Este trabalho propõe um esquema de diferenças finitas compacta de alta ordem para resolver problemas elípticos com coeficientes variáveis em malhas composta. São apresentados a formulação matemática e a dedução do método compacto de quarta ordem aplicado à problemas elípticos bidimensionais, em malha regular e composta. Foi adotado o uso da biblioteca PETSc com os seus pré-condicionadores e métodos numéricos para resolver os sistemas lineares resultantes da discretização do problema. Por fim, testes visando verificar o código foram feitos, utilizando o método de soluções manufaturadas, para mostrar alta eficiência e acurácia do método desenvolvido. / This paper proposes a scheme of compact finite difference higher order for solve elliptic problems with variable coeficients in composite meshes. we present the mathematical formulation and the deduction of the compact method of fourth order applied to two-dimensional elliptic problems in regular and composite mesh . It was adopted using the PETSc library with its pre- conditioners and numerical methods for solving linear systems resulting from discretization of the problem. Finally , tests to verify the code were made using the method of manufactured solutions to show high eficiency and accuracy of the method developed .