Very fine discretizations of differential operators often lead to large, sparse matrices A, where the condition number of A is large. Such ill-conditioning has well known effects on both solving linear systems and eigenvalue computations, and, in general, computing solutions with relative accuracy independent of the condition number is highly desirable. This dissertation is divided into two parts. In the first part, we discuss a method of preconditioning, developed by Ye, which allows solutions of Ax=b to be computed accurately. This, in turn, allows for accurate eigenvalue computations. We then use this method to develop discretizations that yield accurate computations of the smallest eigenvalue of the biharmonic operator across several domains. Numerical results from the various schemes are provided to demonstrate the performance of the methods. In the second part we address the role of the condition number of A in the context of multigrid algorithms. Under various assumptions, we use rigorous Fourier analysis on 2- and 3-grid iteration operators to analyze round off errors in floating point arithmetic. For better understanding of general results, we provide detailed bounds for a particular algorithm applied to the 1-dimensional Poisson equation. Numerical results are provided and compared with those obtained by the schemes discussed in part 1.

Thin and slender structures are widely occurring both in nature and in human creations. Clever geometries of thin structures can produce strong constructions while requiring a minimal amount of material. Computer modeling and analysis of thin and slender structures have their own set of problems, stemming from assumptions made when deriving the governing equations. This thesis deals with the derivation of numerical methods suitable for approximating solutions to problems on thin geometries. It consists of an introduction and four papers. In the first paper we introduce a thread model for use in interactive simulation. Based on a three-dimensional beam model, a corotational approach is used for interactive simulation speeds in combination with adaptive mesh resolution to maintain accuracy. In the second paper we present a family of continuous piecewise linear finite elements for thin plate problems. Patchwise reconstruction of a discontinuous piecewise quadratic deflection field allows us touse a discontinuous Galerkin method for the plate problem. Assuming a criterion on the reconstructions is fulfilled we prove a priori error estimates in energy norm and L2-norm and provide numerical results to support our findings. The third paper deals with the biharmonic equation on a surface embedded in R3. We extend theory and formalism, developed for the approximation of solutions to the Laplace-Beltrami problem on an implicitly defined surface, to also cover the biharmonic problem. A priori error estimates for a continuous/discontinuous Galerkin method is proven in energy norm and L2-norm, and we support the theoretical results by numerical convergence studies for problems on a sphere and on a torus. In the fourth paper we consider finite element modeling of curved beams in R3. We let the geometry of the beam be implicitly defined by a vector distance function. Starting from the three-dimensional equations of linear elasticity, we derive a weak formulation for a linear curved beam expressed in global coordinates. Numerical results from a finite element implementation based on these equations are compared with classical results.

In this thesis, the analysis of the method of fundamental solution(MFS) is expanded for biharmonic equations. The bounds of errors are derived for the traditional and the Almansi's approaches in bounded simply-connected domains. The exponential and the polynomial convergence rates are obtained from highly and finite smooth solutions, respectively. Also the bounds of condition number are derived for the disk domains, to show the exponential growth rates. The analysis in this thesis is the first time to provide the rigor analysis of the CTM for biharmonic equations, and the intrinsic nature of accuracy and stability is similar to that of Laplace's equation. Numerical experiment are carried out for both smooth and singularity problems. The numerical results coincide with the theoretical analysis made. When the particular solutions satisfying the biharmonic equation can be found, the method of particular solutions(MPS) is always superior to MFS, supported by numerical examples. However, if such singular particular solutions near the singular points can not be found, the local refinement of collocation nodes and the greedy adaptive techniques can be used. It seems that the greedy adaptive techniques may provide a better solution for singularity problems. Beside, the numerical solutions by Almansi's approaches are slightly better in accuracy and stability than those by the traditional FS. Hence, the MFS with Almansi's approaches is recommended, due to the simple analysis, which can be obtained directly from the analysis of MFS for Laplace's equation.

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Abstract This dissertation is a combination of four articles on the topic of scattering problems for a biharmonic operator. The operator of interest has two coefficients which may be complex-valued and singular. Each of the articles concerns a different aspect of the problem. Namely, the first article discusses the direct scattering problem in higher dimensions and culminates in a proof of Saito's formula, which yields a uniqueness result for the inverse scattering problem. The second paper is about a backscattering problem in two and three dimensions. We prove that the inverse Born approximation can be used to recover the singularities in the coefficients of the operator. The third article fills in an answer to the question about recovering the complex-valued coefficients in three dimensions that was left open in the second article. The final article studies the inverse scattering problem on the line for a quasi-linear operator. / Tiivistelmä Väitöskirjatyö koostuu neljästä artikkelista, jotka käsittelevät sirontaongelmia biharmoniselle operaattorille. Työn kohteena olevalla operaattorilla on kaksi kerrointa, jotka voivat olla kompleksiarvoisia ja singulaarisia. Kukin artikkeli käsittelee sirontaongelmaa eri näkökulmasta. Ensimmäinen artikkeli koostuu pääasiassa suorasta sirontateoriasta korkeammissa ulottuvuuksissa huipentuen lopulta Saiton kaavan todistukseen, jonka seurauksena saadaan yksikäsitteisyystulos käänteiselle sirontaongelmalle. Toisen artikkelin aiheena on takaisinsirontaongelma kahdessa ja kolmessa ulottuvuudessa. Todistamme, että käänteistä Bornin approksimaatiota voidaan käyttää paikantamaan kertoimien mahdolliset singulariteetit. Kolmas artikkeli vastaa toisessa artikkelissa avoimeksi jääneeseen kysymykseen kompleksiarvoisien kertoimien rekonstruoimisesta kolmessa ulottuvuudessa. Viimeisessä artikkelissa tutkitaan käänteistä sirontaongelmaa kvasilineaariselle operaattorille yhdessä ulottuvuudessa.

Neste trabalho estudamos alguns resultados importantes sobre a teoria das subvariedades bi-harmônicas de espaços homogêneos tridimensionais. Existem três classes de espaços homogêneos tridimensionais simplesmente conexos dependendo da dimensão do grupo de isometrias, que pode ser: 3, 4 ou 6. No caso da dimensão ser 6, M é uma forma espacial; se a dimensão do grupo de isometrias for 4, M é isométrica a: \'H IND. 3\' (grupo de Heisenberg), SU(2) (grupo unitário especial), ~SL(2,R) (revestimento universal do grupo linear especial), ou aos espaços produtos \'S POT. 2\' × R e \'H POT. 2\' × R. Feita exceção para \'H POT. 3\', no caso da dimensão ser 4 ou 6 o espaço homogêneo é localmente isométrico a (uma parte de) \'R POT. 3\', munido de uma métrica que depende de dois parâmetros reais. Tal família de métricas aparece primeiramente no trabalho [3] de L. Bianchi e, mais tarde, nos artigos [14, 35] de É. Cartan e G. Vranceanu, respectivamente. Nesse projeto de mestrado, queremos estudar (essencialmente) resultados de existência e classificação de subvariedades bi-harmônicas nesses espaços, também conhecidos como variedades de Bianchi-Cartan-Vranceanu / In this work we study some important results about the theory of the biharmonic submanifolds of tridimensional homogeneous spaces. There exist three classes of simply connected tridimensional homogeneous spaces depending on the dimension of the group of isometries, which can be: 3, 4 or 6. In the case of dimension 6, M will be a space form; if the dimension of the group of isometries is 4, M will be isometric to: either \'H IND. 3\' (Heisenbergs group), or SU(2) (special unitary group), or ~SL(2,R) (universal recovering of the special linear group), or the product spaces \'S POT. 2\' × R and \'H POT. 2\' × R. Except for \'H POT. 3\', in the case of dimension 4 or 6 the homogeneous space is locally isometric to (a part of) \'R POT. 3\', endowed with a metric that depends on two real parameters. Such family of metrics first appears in the work [3] of L. Bianchi and later in the articles [14, 35] of ´E. Cartan and G. Vranceanu, respectively. In this master thesis, we want to study (essentially) results of existence and classification of bi-harmonic submanifolds in these spaces, also known as Bianchi-Cartan-Vranceanus manifolds

L’objet de cette thèse est l’étude du problème Δ 2uω = fω avec les conditions aux limites Uω = Δ uω = 0, le second membre étant supposé dépendre continûment de ω dans L2(ω), où ω = {(r, θ); 0 < r < 1, 0 < θ < ω} , 0 < ω ≤ π, est une famille de secteurs tronqués du plan. Si ω < π on sait d’après Blum et Rannacher (1980) que la solution de ce problème uω se décompose au voisinage de l’origine en uω = u1,ω + u2,ω + u3,ω, (1) où u1,ω, u2,ω sont les parties singulières de uω et u3,ω la partie régulière. En effet, au voisinage de l’origine u1,ω (resp. u2,ω, u3,ω) est de régularité H1+πω−ǫ (resp. H2+πω−ǫ, H4) pour tout Q > 0, tandis que la solution uπ appartient, au moins au voisinage de l’origine, à l’espace H4(π), où π est le demi-disque supérieur de centre 0 et de rayon r = 1. On voit clairement une résolution de la singularité près de l’angle π dont la description est l’objectif principal de ce travail. Le résultat obtenu est que la décomposition (1) de uω est uniforme par rapport à ω, lorsque ω → π, pour les meilleures topologies possibles pour chacun des termes, et converge terme à terme vers le développement limité de uπ au voisinage de 0. / In this work, we study the family of problems Δ 2uω = fω with boundary conditionuω = Δ uω = 0. There, the second member is assumed to depend smoothly on ω in L2(ω), where ω = {(r, θ); 0 < r < 1, 0 < θ < ω} , 0 < ω ≤ π, is a family of truncated sectors of the plane. If ω < π it is known from Blum et Rannacher (1980) that the solution uω decomposes as uω = u1,ω + u2,ω + u3,ω, (1) where u1,ω, u2,ω are singular and u3,ω is regular. Indeed, near the origin, u1,ω(resp. u2,ω, u3,ω) is of regularity H1+πω−ǫ (resp. H2+πω−ǫ, H4) for every Q > 0, while the solution uπ is, in the neighborhood of the origin again, of regularity H4. One clearly sees a resolution of the singularity near the angle π whose descriptionis the main objective of this work. The obtained result is that there exists a decomposition (1) of uω which is uniform with respect to ω, when ω → π, with the best possible topologies for each term, and which term by term convergestowards the Taylor expansion of uπ near 0.

The buckling of a material subject to stress is a very common phenomenon observed in mechanics. However, the observed buckling of a surface confined hydrogel due to swelling is a unique manifestation of the buckling problem. The reason for buckling is the same in all cases; there is a certain magnitude of force that once exceeded, causes the material to deform itself into a buckling mode. Exactly what that buckling mode is as well as how much force is necessary to cause buckling depends on the material properties. Taking both a finite difference and analytical approach to the problem, it is desired to obtain relationships between the material properties and the predicted buckling modes. These relationships will make it possible for a hydrogel to be designed so that the predicted amount of buckling will occur.

No / In this paper we present a method for generating Bézier surfaces from the boundary information based on a general 4th-order PDE. This is a generalisation of our previous work on harmonic and biharmonic Bézier surfaces whereby we studied the Bézier solutions for Laplace and the standard biharmonic equation, respectively. Here we study the Bézier solutions of the Euler¿Lagrange equation associated with the most general quadratic functional. We show that there is a large class of fourth-order operators for which Bézier solutions exist and hence we show that such operators can be utilised to generate Bézier surfaces from the boundary information. As part of this work we present a general method for generating these Bézier surfaces. Furthermore, we show that some of the existing techniques for boundary based surface design, such as Coons patches and Bloor¿Wilson PDE method, are indeed particular cases of the generalised framework we present here.

Yes