Mixed elliptic problems are characterised by conditions that have a discontinuity on an interface of the boundary of codimension 1. The case of a smooth interface is treated in [3]; the investigation there refers to additional interface conditions and parametrices in standard Sobolev spaces. The present paper studies a necessary structure for the case of interfaces with conical singularities, namely, corner conormal symbols of such operators. These may be interpreted as families of mixed elliptic problems on a manifold with smooth interface. We mainly focus on second order operators and additional interface conditions that are holomorphic in an extra parameter. In particular, for the case of the Zaremba problem we explicitly obtain the number of potential conditions in this context. The inverses of conormal symbols are meromorphic families of pseudo-differential mixed problems referring to a smooth interface. Pointwise they can be computed along the lines [3].

We analyze and compare different techniques to set up the stiffness matrix in the hp-version of the finite element method. The emphasis is on methods for second order elliptic problems posed on meshes including triangular and tetrahedral elements. The polynomial degree may be variable. We present a generalization of the Spectral Galerkin Algorithm of [7], where the shape functions are adapted to the quadrature formula, to the case of triangles/tetrahedra. Additionally, we study on-the-fly matrix-vector multiplications, where merely the matrix-vector multiplication is realized without setting up the stiffness matrix. Numerical studies are included.

The boundary concentrated finite element method is a variant of the hp-version of the finite element method that is particularly suited for the numerical treatment of elliptic boundary value problems with smooth coefficients and low regularity boundary conditions. For this method we present two multilevel preconditioners that lead to preconditioned stiffness matrices with condition numbers that are bounded uniformly in the problem size N. The cost of applying the preconditioners is O(N). Numerical examples illustrate the efficiency of the algorithms.

Η παρούσα διπλωματική αναφέρεται σε ελλειπτικά προβλήματα συνοριακών συνθηκών σε κανονικά πολύγωνα, εστιάζοντας στην αρκετά γενική εξίσωση Helmholtz. Θα εφαρμοσθούν οι γνωστές υπολογιστικές μέθοδοι επίλυσης ελλειπτικών προβλημάτων (όπως η παρεμβολή με τμηματικά κυβικά πολυώνυμα) και θα αναπτυχθούν και μέθοδοι που προκύπτουν από νέες μαθηματικές αναλύσεις του προβλήματος. / In this work we deal with elliptic boundary value problems which are defined in regular polygons. The numerical results presented in the defence are derived using well established methods, such as the finite differemces and the 2d collocation, as well as a new method introduced recently which appears to yield nice results.

In this thesis we consider diverse aspects of existence and correctness of asymptotic solutions to elliptic differential and pseudodifferential equations. We begin our studies with the case of a general elliptic boundary value problem in partial derivatives. A small parameter enters the coefficients of the main equation as well as into the boundary conditions. Such equations have already been investigated satisfactory, but there still exist certain theoretical deficiencies. Our aim is to present the general theory of elliptic problems with a small parameter. For this purpose we examine in detail the case of a bounded domain with a smooth boundary. First of all, we construct formal solutions as power series in the small parameter. Then we examine their asymptotic properties. It suffices to carry out sharp two-sided emph{a priori} estimates for the operators of boundary value problems which are uniform in the small parameter. Such estimates failed to hold in functional spaces used in classical elliptic theory. To circumvent this limitation we exploit norms depending on the small parameter for the functions defined on a bounded domain. Similar norms are widely used in literature, but their properties have not been investigated extensively. Our theoretical investigation shows that the usual elliptic technique can be correctly carried out in these norms. The obtained results also allow one to extend the norms to compact manifolds with boundaries. We complete our investigation by formulating algebraic conditions on the operators and showing their equivalence to the existence of a priori estimates. In the second step, we extend the concept of ellipticity with a small parameter to more general classes of operators. Firstly, we want to compare the difference in asymptotic patterns between the obtained series and expansions for similar differential problems. Therefore we investigate the heat equation in a bounded domain with a small parameter near the time derivative. In this case the characteristics touch the boundary at a finite number of points. It is known that the solutions are not regular in a neighbourhood of such points in advance. We suppose moreover that the boundary at such points can be non-smooth but have cuspidal singularities. We find a formal asymptotic expansion and show that when a set of parameters comes through a threshold value, the expansions fail to be asymptotic. The last part of the work is devoted to general concept of ellipticity with a small parameter. Several theoretical extensions to pseudodifferential operators have already been suggested in previous studies. As a new contribution we involve the analysis on manifolds with edge singularities which allows us to consider wider classes of perturbed elliptic operators. We examine that introduced classes possess a priori estimates of elliptic type. As a further application we demonstrate how developed tools can be used to reduce singularly perturbed problems to regular ones. / In dieser Dissertation betrachten wir verschiedene Aspekte der Existenz und Korrektheit asymptotischer Lösungen für elliptische Differentialgleichungen und Pseudodifferentialgleichungen. Am Anfang betrachtet die Arbeit den Fall eines allgemeinen elliptischen Grenzwertproblems in partiellen Ableitungen. Hierbei hängen die Koeffizienten von einem kleinen Parameter ab. Solche Gleichungen wurden schon reichlich untersucht, aber es gibt immer noch theoretische Lücken. Unser Ziel ist eine allgemeine Theorie elliptischer Operatorklassen mit kleinen Parametern. Zu diesem Zweck untersuchen wir im Detail den Fall eines beschränkten Gebietes mit glattem Rand. Zuerst konstruieren wir formale Lösungen als Potenzreihe einer kleinen Variablen. Weiter untersuchen wir ihre asymptotischen Eigenschaften. Dazu reicht es aus, beidseitige A-Priori Abschätzungen für diejenigen Randwertproblemoperatoren zu bestimmen, die gleichmäßig stetig von den kleinen Parametern abhängen. Solche Abschätzungen gelten nicht in Funktionenräumen, die in der klassischen elliptischen Theorie benutzt werden. Um diese Beschränkungen zu überwinden, nutzen wir Normen abhängig vom kleinen Parameter. Änliche Normen finden sich oft in der Literatur, aber ihre Eigenschaften wurden unzureichend untersucht. Unsere theoretische Forschung zeigt, dass die gewöhnliche elliptische Methode korrekt durchgeführt werden kann. Die erhaltenen Abschätzungen erlauben das Fortsetzen der Normen auf kompakte Mannigfältigkeiten mit Rand. Unsere Forschung wird mit algebraischen Bedingungen für die Operatoren abgeschlossen. Wir zeigen, dass diese Bedingungen äquivalent zu der Existenz der A-Priori-Abschätzungen sind. Im zweiten Schritt erweitern wir das Konzept der Elliptizität mit kleinen Parametern zu allgemeineren Operatorklassen. Zuerst wollen wir den Unterschied in asymptotischen Mustern zwischen der erhaltenen Reihe und Lösungen ähnlicher Probleme untersuchen. Deshalb untersuchen wir die Wärmeleitungsgleichung in einem beschränkten Gebiet mit einem kleinen Parameter in der Zeitableitung. In diesem Fall tangiert der Rand die Charakteristik endlich oft. Es ist bekannt, dass die Lösungen unregulär im Allgemeinen in Umgebungen solcher Stellen sind. Wir nehmen an, dass der Rand an solchen Stellen nicht glätt sein kann und kaspydalische Singularitäten hat. Wir haben eine formale asymptotische Zerlegung gefunden und einen Schwellenwert gezeigt, sodass die asymptotische Eigenschaft der Reihe nicht mehr gilt, wenn der Randparameter diesen Schwellenwert übersteigt. Der letze Teil der Arbeit führt ein allgemeines Konzept der Elliptizit"at mit einem kleinen Parameter ein. Mehrere theoretische Erweiterungen auf Pseudodifferentialoperatoren wurden schon in früheren Studien vorgeschlagen. Als neuen Beitrag wenden wir die Analysis auf Manigfältigkeiten mit Kantensingularitäten an. Dies lässt es zu, allgemeinere gestörte Operatorklassen zu betrachten. Wir beobachten, dass die eingef"uhrten Klassen A-Priori-Abschätzungen elliptischer Gestalt haben. Als weitere Anwendung demonstrieren wir, wie die entwickelten Mittel zum Reduzieren singular gestörter Probleme zu regulären Fällen benutzt werden können.

In spectral methods for numerical PDEs, when the solutions are periodical, the Fourier functions may be used. However, when the solutions are non-periodical, the Legendre and Chebyshev polynomials are recommended, reported in many papers and books. There seems to exist few reports for the study of non-periodical solutions by spectral Fourier methods under the Dirichlet conditions and other boundary conditions. In this paper, we will explore the spectral Fourier methods(SFM) and collocation Fourier methods(CFM) for elliptic and eigenvalue problems. The CFM is simple and easy for computation, thus for saving a great deal of the CPU time. The collocation Fourier methods (CFM) can be regarded as the spectral Fourier methods (SFM) partly with the trapezoidal rule. Furthermore, the error bounds are derived for both the CFM and the SFM. When there exist no errors for the trapezoidal rule, the accuracy of the solutions from the CFM is as accurate as the spectral method using Legendre and Chebyshev polynomials. However, once there exists the truncation errors of the trapezoidal rule, the errors of the elliptic solutions and the leading eigenvalues the CFM are reduced to O(h^2), where h is the mesh length of uniform collocation grids, which are just equivalent to those by the linear elements and the finite difference method (FDM). The O(h^2) and even the superconvergence O(h4) are found numerically. The traditional condition number of the CFM is O(N^2), which is smaller than O(N^3) and O(N^4) of the collocation spectral methods using the Legendre and Chebyshev polynomials. Also the effective condition number is only O(1). Numerical experiments are reported for 1D elliptic and eigenvalue problems, to support the analysis made. The simplicity of algorithms and the promising numerical computation with O(h^4) may grant the CFM to be competent in application in numerical physics, chemistry, engineering, etc., see [7].

Submitted by Aelson Maciera (aelsoncm@terra.com.br) on 2017-06-05T20:04:17Z No. of bitstreams: 1 TeseEJH.pdf: 21600349 bytes, checksum: 9be6866d2a15e1ea7aa6325d99e7fa4c (MD5) / Approved for entry into archive by Ronildo Prado (ronisp@ufscar.br) on 2017-06-06T13:12:02Z (GMT) No. of bitstreams: 1 TeseEJH.pdf: 21600349 bytes, checksum: 9be6866d2a15e1ea7aa6325d99e7fa4c (MD5) / Approved for entry into archive by Ronildo Prado (ronisp@ufscar.br) on 2017-06-06T13:12:09Z (GMT) No. of bitstreams: 1 TeseEJH.pdf: 21600349 bytes, checksum: 9be6866d2a15e1ea7aa6325d99e7fa4c (MD5) / Made available in DSpace on 2017-06-06T13:20:52Z (GMT). No. of bitstreams: 1 TeseEJH.pdf: 21600349 bytes, checksum: 9be6866d2a15e1ea7aa6325d99e7fa4c (MD5) Previous issue date: 2016-12-05 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) / We study the existence and multiplicity of nontrivial solutions for two classes of elliptic problems. The first problem covers a general class of operators with variables exponents where the nonlinearitv has subcritical growth. The second problem is a nonlocal elliptic problem where the nonlinearitv has critical growth. ... continua / Neste trabalho estudamos a existência e multiplicidade de soluções não triviais para duas classes de problemas elípticos. O primeiro problema elíptico que estudamos abrange uma classe geral de operadores com expoentes variáveis onde a não linearidade possui crescimento subcrítico. O segundo problema trata de uma equação não local envolvendo uma ampla classe de operadores onde a não linearidade possui crescimento sublinear/superlinear, mais um termo com crescimento crítico. ... continua

Submitted by Renata Lopes (renatasil82@gmail.com) on 2017-05-26T17:51:05Z No. of bitstreams: 1 sandramachadodesouzalima.pdf: 680308 bytes, checksum: 1b724b63bb7a52093f6e1411a716269f (MD5) / Approved for entry into archive by Adriana Oliveira (adriana.oliveira@ufjf.edu.br) on 2017-05-29T18:54:12Z (GMT) No. of bitstreams: 1 sandramachadodesouzalima.pdf: 680308 bytes, checksum: 1b724b63bb7a52093f6e1411a716269f (MD5) / Made available in DSpace on 2017-05-29T18:54:12Z (GMT). No. of bitstreams: 1 sandramachadodesouzalima.pdf: 680308 bytes, checksum: 1b724b63bb7a52093f6e1411a716269f (MD5) Previous issue date: 2014-07-03 / FAPEMIG - Fundação de Amparo à Pesquisa do Estado de Minas Gerais / A variedade de Nehari para a equação −∆u(x) = λa(x)u(x)q + b(x)u(x)p, com x ∈ Ω, junto com a condição de fronteira de Dirichlet é investigada no caso em que a(x) = 1, λ ∈R, q = 1 e 0 < p < 1, e também no caso em que λ > 0 e 0 < q < 1 < p < 2∗−1. Explorando a relação entre a variedade de Nehari e a aplicação fibração ( isto é, aplicações da forma t → J(tu) onde J é o funcional de Euler associado ao problema em questão), iremos discutir a existência e multiplicidade de soluções não negativas. / The Nehari Manifold for the equation −∆u(x) = λa(x)u(x)q + b(x)u(x)p, for x ∈ Ω together with Dirichlet boundary conditions is investigated in which case a(x) = 1, λ ∈R, q = 1 and 0 < p < 1, and also in the case that λ > 0 and 0 < q < 1 < p < 2∗−1. Exploring the relationship between the Nehari manifold and fibering maps (i.e., maps of the form t → J(tu) where J is the Euler functional associated to the above equation), we will discuss the existence and multiplicity of non negative solutions.

We analyze and compare different techniques to set up the stiffness matrix in the hp-version of the finite element method. The emphasis is on methods for second order elliptic problems posed on meshes including triangular and tetrahedral elements. The polynomial degree may be variable. We present a generalization of the Spectral Galerkin Algorithm of [7], where the shape functions are adapted to the quadrature formula, to the case of triangles/tetrahedra. Additionally, we study on-the-fly matrix-vector multiplications, where merely the matrix-vector multiplication is realized without setting up the stiffness matrix. Numerical studies are included.

The boundary concentrated finite element method is a variant of the hp-version of the finite element method that is particularly suited for the numerical treatment of elliptic boundary value problems with smooth coefficients and low regularity boundary conditions. For this method we present two multilevel preconditioners that lead to preconditioned stiffness matrices with condition numbers that are bounded uniformly in the problem size N. The cost of applying the preconditioners is O(N). Numerical examples illustrate the efficiency of the algorithms.