Yes / This paper extends the PDE method of surface generation. The governing partial differential equation is generalised to sixth order to increase its flexibility. The PDE is solved analytically, even in the case of general boundary conditions, making the method fast. The boundary conditions, which control the surface shape, are specified interactively, allowing intuitive manipulation of generic shapes. A compact user interface is presented which makes use of direct manipulation and other techniques for 3D interaction.

<p>The present thesis is devoted to the homogenization of certain elliptic and parabolic partial differential equations by means of appropriate generalizations of the notion of two-scale convergence. Since homogenization is defined in terms of H-convergence, we desire to find the H-limits of sequences of periodic monotone parabolic operators with two spatial scales and an arbitrary number of temporal scales and the H-limits of sequences of two-dimensional possibly non-periodic linear elliptic operators by utilizing the theories for evolution-multiscale convergence and λ-scale convergence, respectively, which are generalizations of the classical two-scale convergence mode and custom-made to treat homogenization problems of the prescribed kinds. Concerning the multiscaled parabolic problems, we find that the result of the homogenization depends on the behavior of the temporal scale functions. The temporal scale functions considered in the thesis may, in the sense explained in the text, be slow or rapid and in resonance or not in resonance with respect to the spatial scale function. The homogenization for the possibly non-periodic elliptic problems gives the same result as for the corresponding periodic problems but with the exception that the local gradient operator is everywhere substituted by a differential operator consisting of a product of the local gradient operator and matrix describing the geometry and which depends, effectively, parametrically on the global variable.</p>

The present thesis is devoted to the homogenization of certain elliptic and parabolic partial differential equations by means of appropriate generalizations of the notion of two-scale convergence. Since homogenization is defined in terms of H-convergence, we desire to find the H-limits of sequences of periodic monotone parabolic operators with two spatial scales and an arbitrary number of temporal scales and the H-limits of sequences of two-dimensional possibly non-periodic linear elliptic operators by utilizing the theories for evolution-multiscale convergence and λ-scale convergence, respectively, which are generalizations of the classical two-scale convergence mode and custom-made to treat homogenization problems of the prescribed kinds. Concerning the multiscaled parabolic problems, we find that the result of the homogenization depends on the behavior of the temporal scale functions. The temporal scale functions considered in the thesis may, in the sense explained in the text, be slow or rapid and in resonance or not in resonance with respect to the spatial scale function. The homogenization for the possibly non-periodic elliptic problems gives the same result as for the corresponding periodic problems but with the exception that the local gradient operator is everywhere substituted by a differential operator consisting of a product of the local gradient operator and matrix describing the geometry and which depends, effectively, parametrically on the global variable.

Orientador: Djairo Guedes de Figueiredo / Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Matemática, Estatística e Computação Científica / Made available in DSpace on 2018-08-18T23:11:17Z (GMT). No. of bitstreams: 1 Monteiro_Evandro_D.pdf: 681089 bytes, checksum: 5ec4729a2d7b386329193adf424f6b42 (MD5) Previous issue date: 2011 / Resumo: O resumo, na íntegra, poderá ser visualizado no texto completo da tese digital / Abstract: The complete abstract is available with the full electronic digital thesis or dissertations / Doutorado / Matematica / Doutor em Matemática

Orientador: Francisco Odair Vieira de Paiva / Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Matemática, Estatística e Computação Científica / Made available in DSpace on 2018-08-19T21:33:51Z (GMT). No. of bitstreams: 1 Cardoso_JoseAndersonValenca_D.pdf: 1655484 bytes, checksum: 6e4f6872240f3317db759e94789f5d34 (MD5) Previous issue date: 2012 / Resumo: Neste trabalho consideramos uma classe de Sistemas Elípticos Hamiltonianos. Esta classe de sistemas surge como modelo natural em áreas como Física e Biologia. Estudamos casos que envolvem crescimento crítico, arbitrário e crítico perturbado e analisamos questões relacionadas a existência, multiplicidade e propriedades de soluções. Os resultados são obtidos com o uso de métodos variacionais, a exemplo dos teoremas de min-max, aliados as propriedades das funções com simetria radial e ao princípio de concentração de compacidade / Abstract: In this work, we consider a class of Hamiltonian Elliptic Systems. This class of systems arise as a natural model in many areas such as Physics and Biology. We studied cases involving critical growth, arbitrary growth and perturbed critical growth and we also investigated questions related to the existence, multiplicity and properties of solutions. The results are obtained by using a variational approach, for instance, min-max theorems, combined with properties of radially symmetric functions and the concentration-compactness principle / Doutorado / Matematica / Doutor em Matemática

No description available.

Thesis (MSc)--Stellenbosch University, 2011. / ENGLISH ABSTRACT: In several numerical approaches to PDEs shifted linear systems of the form (zI - A)x = b, need to be solved for several values of the complex scalar z. Often, these linear systems are large and sparse. This thesis investigates efficient numerical methods for these systems that arise from a contour integral approximation to PDEs and compares these methods with direct solvers. In the first part, we present three model PDEs and discuss numerical approaches to solve them. We use the first problem to demonstrate computations with a dense matrix, the second problem to demonstrate computations with a sparse symmetric matrix and the third problem for a sparse but nonsymmetric matrix. To solve the model PDEs numerically we apply two space discrerization methods, namely the finite difference method and the Chebyshev collocation method. The contour integral method mentioned above is used to integrate with respect to the time variable. In the second part, we study a Hessenberg reduction method for solving shifted linear systems with a dense matrix and present numerical comparison of it with the built-in direct linear system solver in SciPy. Since both are direct methods, in the absence of roundoff errors, they give the same result. However, we find that the Hessenberg reduction method is more efficient in CPU-time than the direct solver. As application we solve a one-dimensional version of the heat equation. In the third part, we present efficient techniques for solving shifted systems with a sparse matrix by Krylov subspace methods. Because of their shift-invariance property, the Krylov methods allow one to obtain approximate solutions for all values of the parameter, by generating a single approximation space. Krylov methods applied to the linear systems are generally slowly convergent and hence preconditioning is necessary to improve the convergence. The use of shift-invert preconditioning is discussed and numerical comparisons with a direct sparse solver are presented. As an application we solve a two-dimensional version of the heat equation with and without a convection term. Our numerical experiments show that the preconditioned Krylov methods are efficient in both computational time and memory space as compared to the direct sparse solver. / AFRIKAANSE OPSOMMING: In verskeie numeriese metodes vir PDVs moet geskuifde lineêre stelsels van die vorm (zI − A)x = b, opgelos word vir verskeie waardes van die komplekse skalaar z. Hierdie stelsels is dikwels groot en yl. Hierdie tesis ondersoek numeriese metodes vir sulke stelsels wat voorkom in kontoerintegraalbenaderings vir PDVs en vergelyk hierdie metodes met direkte metodes vir oplossing. In die eerste gedeelte beskou ons drie model PDVs en bespreek numeriese benaderings om hulle op te los. Die eerste probleem word gebruik om berekenings met ’n vol matriks te demonstreer, die tweede probleem word gebruik om berekenings met yl, simmetriese matrikse te demonstreer en die derde probleem vir yl, onsimmetriese matrikse. Om die model PDVs numeries op te los beskou ons twee ruimte-diskretisasie metodes, naamlik die eindige-verskilmetode en die Chebyshev kollokasie-metode. Die kontoerintegraalmetode waarna hierbo verwys is word gebruik om met betrekking tot die tydveranderlike te integreer. In die tweede gedeelte bestudeer ons ’n Hessenberg ontbindingsmetode om geskuifde lineêre stelsels met ’n vol matriks op te los, en ons rapporteer numeriese vergelykings daarvan met die ingeboude direkte oplosser vir lineêre stelsels in SciPy. Aangesien beide metodes direk is lewer hulle dieselfde resultate in die afwesigheid van afrondingsfoute. Ons het egter bevind dat die Hessenberg ontbindingsmetode meer effektief is in terme van rekenaartyd in vergelyking met die direkte oplosser. As toepassing los ons ’n een-dimensionele weergawe van die hittevergelyking op. In die derde gedeelte beskou ons effektiewe tegnieke om geskuifde stelsels met ’n yl matriks op te los, met Krylov subruimte-metodes. As gevolg van hul skuifinvariansie eienskap, laat die Krylov metodes mens toe om benaderde oplossings te verkry vir alle waardes van die parameter, deur slegs een benaderingsruimte voort te bring. Krylov metodes toegepas op lineêre stelsels is in die algemeen stadig konvergerend, en gevolglik is prekondisionering nodig om die konvergensie te verbeter. Die gebruik van prekondisionering gebasseer op skuif-en-omkeer word bespreek en numeriese vergelykings met direkte oplossers word aangebied. As toepassing los ons ’n twee-dimensionele weergawe van die hittevergelyking op, met ’n konveksie term en daarsonder. Ons numeriese eksperimente dui aan dat die Krylov metodes met prekondisionering effektief is, beide in terme van berekeningstyd en rekenaargeheue, in vergelyking met die direkte metodes.

The behaviour of free-surface gravity waves on small Froude number fluid flow past some obstacle cannot be determined using ordinary asymptotic power series methods, as the amplitude of the waves is exponentially small. An exponential asymptotic method is used by Chapman and Vanden-Broeck (2006) to consider the problem of two-dimensional, steady flow past a submerged obstacle in the small Froude number limit, finding that a steady downstream wavetrainis switched on rapidly across a curve known as a Stokes line. Here, equivalent wavetrains on three-dimensional and unsteady flow configurations are considered, and Stokes switching causedby the interaction between exponentially small free-surface components is shown to play an important role in both cases. The behaviour of free-surface gravity waves is introduced by considering the problem of steady free-surface flow due to a line source. A steady wavetrain is shown to exist in the far field, and the behaviour of these waves is compared to existing numerical results. The problem of unsteady flow over a step is subsequently investigated, with the flow behaviour formulated in terms of Lagrangian coordinates so that the position of the free surface is fixed. Initially, the problem is linearized in the step-height, and the steady wavetrain is shown to spread downstream over time. The position of the wavefront is determined by considering the full Stokes structure present in the problem. The equivalent fully-nonlinear problem is then considered, with the position of the Stokes lines, and hence the wavefront, being determined numerically. Finally, linearized three-dimensional free-surface flow past an obstacle is considered in both the steady and unsteady case. The surface is shown to contain downstream longitudinal and transverse waves. These waves are shown to propagate downstream in the unsteady case, with the position of the wavefront again determined by considering the full Stokes structure of the problem.

Mémoire numérisé par la Direction des bibliothèques de l'Université de Montréal.

Einstein's field equations are a system of ten partial differential equations. For a special class of spacetimes known as Gowdy spacetimes, the number of equations is reduced due to additional structure of two dimensional isometry groups with mutually orthogonal Killing vectors. In this thesis, we focus on a particular model of Gowdy spacetimes known as the polarized T3 model, and provide an explicit solution to Einstein's equations.