The Lower Bounds for Eigenvalues of Elliptic Operators --By Nonconforming Finite Element Methods

The aim of the paper is to introduce a new systematic method that can produce lower bounds for eigenvalues. The main idea is to use nonconforming finite element methods. The general conclusion herein is that if local approximation properties of nonconforming finite element spaces $V_h$ are better than global continuity properties of $V_h$, corresponding methods will produce lower bounds for eigenvalues. More precisely, under three conditions on continuity and approximation properties of nonconforming finite element spaces we first show abstract error estimates of approximate eigenvalues and eigenfunctions. Subsequently, we propose one more condition and prove that it is sufficient to guarantee nonconforming finite element methods to produce lower bounds for eigenvalues of symmetric elliptic operators. As one application, we show that this condition hold for most nonconforming elements in literature. As another important application, this condition provides a guidance to modify known nonconforming elements in literature and to propose new nonconforming elements. In fact, we enrich locally the Crouzeix-Raviart element such that the new element satisfies the condition; we propose a new nonconforming element for second order elliptic operators and prove that it will yield lower bounds for eigenvalues. Finally, we prove the saturation condition for most nonconforming elements.Comment: 24 page

The MHM Method for Elasticity on Polytopal Meshes

The multiscale hybrid-mixed (MHM) method consists of a multi-level strategy to approximate the solution of boundary value problems with heterogeneous coefficients. In this context, we propose a new family of finite elements for the linear elasticity equation defined on coarse polytopal partitions of the domain. The finite elements rely on face degrees of freedom associated with multiscale bases obtained from local Neumann problems with polynomial interpolations on faces. We establish sufficient conditions on the fine-scale interpolations such that the MHM method is well-posed. Also, discrete traction stays in local equilibrium with external forces. We show by means of a multi-level analysis that the MHM method achieves optimal convergence under local regularity conditions without refining the coarse partition. The upshot is that the Poincar\'e and Korn's inequalities do not degenerate, and then convergence arises on general meshes. We employ two- and three-dimensional numerical tests to assess theoretical results and to verify the robustness of the method through a multi-layer media case. Also, we address computational aspects of the underlying parallel algorithm associated with different configurations of the MHM method; our aim is to find the best compromise between execution time and memory allocation to achieve a given error threshold

Synthesis and evaluation of geometric textures

Two-dimensional geometric textures are the geometric analogues of raster (pixel-based) textures and consist of planar distributions of discrete shapes with an inherent structure. These textures have many potential applications in art, computer graphics, and cartography. Synthesizing large textures by hand is generally a tedious task. In raster-based synthesis, many algorithms have been developed to limit the amount of manual effort required. These algorithms take in a small example as a reference and produce larger similar textures using a wide range of approaches. Recently, an increasing number of example-based geometric synthesis algorithms have been proposed. I refer to them in this dissertation as Geometric Texture Synthesis (GTS) algorithms. Analogous to their raster-based counterparts, GTS algorithms synthesize arrangements that ought to be judged by human viewers as “similar” to the example inputs. However, an absence of conventional evaluation procedures in current attempts demands an inquiry into the visual significance of synthesized results. In this dissertation, I present an investigation into GTS and report on my findings from three projects. I start by offering initial steps towards grounding texture synthesis techniques more firmly with our understanding of visual perception through two psychophysical studies. My observations throughout these studies result in important visual cues used by people when generating and/or comparing similarity of geometric arrangements as well a set of strategies adopted by participants when generating arrangements. Based on one of the generation strategies devised in these studies I develop a new geometric synthesis algorithm that uses a tile-based approach to generate arrangements. Textures synthesized by this algorithm are comparable to the state of the art in GTS and provide an additional reference in subsequent evaluations. To conduct effective evaluations of GTS, I start by collecting a set of representative examples, use them to acquire arrangements from multiple sources, and then gather them into a dataset that acts as a standard for the GTS research community. I then utilize this dataset in a second set of psychophysical studies that define an effective methodology for comparing current and future geometric synthesis algorithms

Classical solutions for stabilized periodic Hele-Shaw flows with a free surface

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Isogeometric Analysis for Reduced Fluid-Structure Interaction Models in Haemodynamic Applications

Isogeometric analysis (IGA) is a computational methodology recently developed to numerically approximate Partial Differential Equation (PDEs). It is based on the isogeometric paradigm, for which the same basis functions used to represent the geometry are then used to approximate the unknown solution of the PDEs. In the case in which Non-Uniform Rational B-Splines (NURBS) are used as basis functions, their mathematical properties lead to appreciable benefits for the numerical approximation of PDEs, especially for high order PDEs in the standard Galerkin formulation. In this framework, we propose an a priori error estimate, extending existing results limited to second order PDEs. The improvements in both accuracy and efficiency of IGA compared to Finite Element Analysis (FEA), encourage the use of this methodology in the haemodynamic applications. In fact, the simulation of blood flow in arteries requires the numerical approximation of Fluid-Structure Interaction (FSI) problems. In order to account for the deformability of the vessel, the Navier-Stokes equations representing the blood flows, are coupled with structural models describing the mechanical response of the arterial wall. However, the FSI models are complex from both the mathematical and the numerical points of view, leading to high computational costs during the simulations. With the aim of reducing the complexity of the problem and the computational costs of the simulations, reduced FSI models can be considered. A first simplification, based on the assumption of a thin arterial wall structure, consists in considering shell models to describe the mechanical properties of the arterial walls. Moreover, by means of the additional kinematic condition (continuity of velocities) and dynamic condition (balance of contact forces), the structural problem can be rewritten as generalized boundary condition for the fluid problem. This results in a generalized Navier-Stokes problem which can be expressed only in terms of the primitive variables of the fluid equations (velocity and pressure) and in a fixed computational domain. As a consequence, the computational costs of the numerical simulations are significantly reduced. On the other side, the generalized boundary conditions associated to the reduced FSI model could involve high order derivatives, which need to be suitably approximated. With this respect, IGA allows an accurate, straightforward and efficient numerical approximation of the generalized Navier-Stokes equations characterizing the reduced FSI problem. In this work we consider the numerical approximation of reduced FSI models by means of IGA, for which we discuss the numerical results obtained in Haemodynamic applications

Penalty-free Nitsche method for interface problems in computational mechanics

Nitsche’s method is a penalty-based method to enforce weakly the boundary conditions in the finite element method. In this thesis, we consider a penalty-free version of Nitsche’s method, we prove its stability and convergence in various frameworks. The idea of the penalty-free method comes from the nonsymmetric version of the Nitsche’s method where the penalty parameter has been set to zero; it can be seen as a Lagrange multiplier method, where the Lagrange multiplier has been replaced by the boundary fluxes of the discrete elliptic operator. The main observation is that although coercivity fails, inf-sup stability can be proven. The study focuses on compressible and incompressible elasticity. An unfitted framework is considered when the computational mesh does not fit with the physical domain (fictitious domain method). The penalty-free Nitsche’s method is also used to enforce the coupling for interface problems when the mesh fits the interface (nonconforming domain decomposition) or not (unfitted domain decomposition). Fluid structure interaction is also investigated, a new fully discrete implicit scheme is introduced

Spectral and High Order Methods for Partial Differential Equations ICOSAHOM 2018

This open access book features a selection of high-quality papers from the presentations at the International Conference on Spectral and High-Order Methods 2018, offering an overview of the depth and breadth of the activities within this important research area. The carefully reviewed papers provide a snapshot of the state of the art, while the extensive bibliography helps initiate new research directions

Numerical Methods for Deterministic and Stochastic Phase Field Models of Phase Transition and Related Geometric Flows

This dissertation consists of three integral parts with each part focusing on numerical approximations of several partial differential equations (PDEs). The goals of each part are to design, to analyze and to implement continuous or discontinuous Galerkin finite element methods for the underlying PDE problem. Part One studies discontinuous Galerkin (DG) approximations of two phase field models, namely, the Allen-Cahn and Cahn-Hilliard equations, and their related curvature-driven geometric problems, namely, the mean curvature flow and the Hele-Shaw flow. We derive two discrete spectrum estimates, which play an important role in proving the sharper error estimates which only depend on a negative power of the singular perturbation parameter ε [epsilon] instead of an exponential power. It is also proved that the zero level sets of the numerical solutions of the Allen-Cahn equation and the Cahn-Hilliard equation approximate the mean curvature flow and the Hele-Shaw flow respectively. Numerical experiments are carried out to verify the theoretical results and to compare the zero level sets of the numerical solutions and the geometric motions. Part Two focuses on finite element approximations of stochastic geometric PDEs including the phase field formulation of a stochastic mean curvature flow and the level set formulation of the stochastic mean curvature flow. Both formulations give PDEs with gradient-type multiplicative noises. We establish PDE energy laws and the Hölder [Holder] continuity in time for the exact solutions. Moreover, optimal error estimates are derived, and various numerical experiments are carried out to study the interplay of the geometric evolution and gradient-type noises. Part Three studies finite element methods for a quasi-static model of poroelasticity, which is a fluid-solid interaction multiphysics system at pore scale. We reformulate the original multiphysics system into a new system which explicitly reveals the diffusion process and has a built-in mechanism to overcome the locking phenomenon . Fully discrete finite element methods are proposed for approximating the new system. We derive a discrete energy law and optimal error estimates for our finite element methods. Numerical experiments are also provided to verify the theoretical results and to confirm that the locking phenomenon has indeed been overcome