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HigherDegree Immersed Finite Elements for SecondOrder Elliptic Interface ProblemsBen Romdhane, Mohamed 16 September 2011 (has links)
A wide range of applications involve interface problems. In most of the cases, mathematical modeling of these interface problems leads to partial differential equations with nonsmooth or discontinuous inputs and solutions, especially across material interfaces. Different numerical methods have been developed to solve these kinds of problems and handle the nonsmooth behavior of the input data and/or the solution across the interface. The main focus of our work is the immersed finite element method to obtain optimal numerical solutions for interface problems.
In this thesis, we present piecewise quadratic immersed finite element (IFE) spaces that are used with an immersed finite element (IFE) method with interior penalty (IP) for solving twodimensional secondorder elliptic interface problems without requiring the mesh to be aligned with the material interfaces. An analysis of the constructed IFE spaces and their dimensions is presented. Shape functions of Lagrange and hierarchical types are constructed for these spaces, and a proof for the existence is established. The interpolation errors in the proposed piecewise quadratic spaces yield optimal <i>O</i>(h³) and <i>O</i>(h²) convergence rates, respectively, in the L² and broken H¹ norms under mesh refinement. Furthermore, numerical results are presented to validate our theory and show the optimality of our quadratic IFE method.
Our approach in this thesis is, first, to establish a theory for the simplified case of a linear interface. After that, we extend the framework to quadratic interfaces. We, then, describe a general procedure for handling arbitrary interfaces occurring in real physical practical applications and present computational examples showing the optimality of the proposed method. Furthermore, we investigate a general procedure for extending our quadratic IFE spaces to <i>p</i>th degree and construct hierarchical shape functions for <i>p</i>=3. / Ph. D.

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Design, Analysis, and Application of Immersed Finite Element MethodsGuo, Ruchi 19 June 2019 (has links)
This dissertation consists of three studies of immersed finite element (IFE) methods for inter face problems related to partial differential equations (PDEs) with discontinuous coefficients. These three topics together form a continuation of the research in IFE method including the extension to elasticity systems, new breakthroughs to higher degree IFE methods, and its application to inverse problems.
First, we extend the current construction and analysis approach of IFE methods in the literature for scalar elliptic equations to elasticity systems in the vector format. In particular, we construct a group of lowdegree IFE functions formed by linear, bilinear, and rotated Q1 polynomials to weakly satisfy the jump conditions of elasticity interface problems. Then we analyze the trace inequalities of these IFE functions and the approximation capabilities of the resulted IFE spaces. Based on these preparations, we develop a partially penalized IFE (PPIFE) scheme and prove its optimal convergence rates.
Secondly, we discuss the limitations of the current approaches of IFE methods when we try to extend them to higher degree IFE methods. Then we develop a new framework to construct and analyze arbitrary pth degree IFE methods. In this framework, each IFE function is the extension of a pth degree polynomial from one subelement to the whole interface element by solving a local Cauchy problem on interface elements in which the jump conditions across the interface are employed as the boundary conditions. All the components in the analysis, including existence of IFE functions, the optimal approximation capabilities and the trace inequalities, are all reduced to key properties of the related discrete extension operator. We employ these results to show the optimal convergence of a discontinuous Galerkin IFE (DGIFE) method.
In the last part, we apply the linear IFE methods in the literature together with the shape optimization technique to solve a group of interface inverse problems. In this algorithm, both the governing PDEs and the objective functional for interface inverse problems are discretized optimally by the IFE method regardless of the location of the interface in a chosen mesh. We derive the formulas for the gradients of the objective function in the optimization problem which can be implemented efficiently in the IFE framework through a discrete adjoint method. We demonstrate the properties of the proposed algorithm by applying it to three representative applications. / Doctor of Philosophy / Interface problems arise from many science and engineering applications modeling the transmission of some physical quantities between multiple materials. Mathematically, these multiple materials in general are modeled by partial differential equations (PDEs) with discontinuous parameters, which poses challenges to developing efficient and reliable numerical methods and the related theoretical error analysis. The main contributions of this dissertation is on the development of a special finite element method, the so called immersed finite element (IFE) method, to solve the interface problems on a mesh independent of the interface geometry which can be advantageous especially when the interface is moving. Specifically, this dissertation consists of three projects of IFE methods: elasticity interface problems, higherorder IFE methods and interface inverse problems, including their design, analysis, and application.

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Application of harmonic coordinates to 2D interface problems on regular gridsJanuary 2012 (has links)
Finite difference and finite element methods exhibit first order convergence when applied to static interface problems where the grid and interface are not aligned. Although modified and unstructured grid methods would address the issue of misalignment for finite elements, application to large models of stratified media, such as those encountered in exploration geophysics, may require not only manual mesh manipulation but also more degrees of freedom than are ultimately necessary to resolve the solution. Instead using fitted or otherwise modified grids, this thesis details an improvement to an existing upscaling method that incorporates finescale variations of material properties by composing standard piecewise linear basis functions with a specific type of harmonic map. This technique requires that the problem domain be discretized using two meshes: one fine mesh where the harmonic map is computed to resolve finescale structures, and a coarse mesh where the solution to the problem is approximated. The implementation of this method in the literature restricts these composite basis functions to triangular elements in 2D leading to a nonconforming finite element method and suboptimal convergence. However, the support of these basis functions in harmonic coordinates is triangular. I present a meshmesh intersection algorithm that exploits this alternative representation to determine the true support of the composite basis functions in terms of the fine mesh. The result is a conforming, highresolution finite element basis that is associated with the original coarse mesh nodes. Leveraging this fine scale information, I develop a new finite element matrix assembly algorithm. Knowing the shape of the basis support leads naturally to an integration method for computing the finite element matrix entries that is exact up to the accuracy of the harmonic map approximation. This new conforming method is shown to improve the accuracy of solutions to elliptic PDE with discontinuous coefficients on coarse, regular grids.

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Immersed Discontinuous Galerkin Methods for Acoustic Wave Propagation in Inhomogeneous MediaMoon, Kihyo 03 May 2016 (has links)
We present immersed discontinuous Galerkin finite element methods for one and two dimensional acoustic wave propagation problems in inhomogeneous media where elements are allowed to be cut by the material interface. The proposed methods use the standard discontinuous Galerkin finite element formulation with polynomial approximation on elements that contain one fluid while on interface elements containing more than one fluid they use speciallybuilt piecewise polynomial shape functions that satisfy appropriate interface jump conditions. The finite element spaces on interface elements satisfy physical interface conditions from the acoustic problem in addition to extended conditions derived from the system of partial differential equations. Additional curlfree and consistency conditions are added to generate bilinear and biquadratic piecewise shape functions for two dimensional problems. We established the existence and uniqueness of one dimensional immersed finite element shape functions and existence of two dimensional bilinear immersed finite element shape functions for the velocity.
The proposed methods are tested on one dimensional problems and are extended to two dimensional problems where the problem is defined on a domain split by an interface into two different media. Our methods exhibit optimal $O(h^{p+1})$ convergence rates for one and two dimensional problems. However it is observed that one of the proposed methods is not stable for two dimensional interface problems with high contrast media such as water/air. We performed an analysis to prove that our immersed PetrovGalerkin method is stable for interface problems with high jumps across the interface. Local timestepping and parallel algorithms are used to speed up computation.
Several realistic interface problems such as ether/glycerol, water/methylalcohol and water/air with a circular interface are solved to show the stability and robustness of our methods. / Ph. D.

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Nonconforming Immersed Finite Element Methods for Interface ProblemsZhang, Xu 04 May 2013 (has links)
In science and engineering, many simulations are carried out over domains consisting of multiple materials separated by curves/surfaces. If partial differential equations (PDEs) are used to model these simulations, it usually leads to the socalled interface problems of PDEs whose coefficients are discontinuous. In this dissertation, we consider nonconforming immersed "nite element (IFE) methods and error analysis for interface problems.
We "first consider the second order elliptic interface problem with a discontinuous diffusion coefficient. We propose new IFE spaces based on the nonconforming rotated Q1 "finite elements on Cartesian meshes. The degrees of freedom of these IFE spaces are determined by midpoint values or average integral values on edges. We investigate fundamental properties of these IFE spaces, such as unisolvency and partition of unity, and extend wellknown trace inequalities and inverse inequalities to these IFE functions. Through interpolation error analysis, we prove that these IFE spaces have optimal approximation capabilities.
We use these IFE spaces to develop partially penalized Galerkin (PPG) IFE schemes whose bilinear forms contain penalty terms over interface edges. Error estimation is carried out for these IFE schemes. We prove that the PPG schemes with IFE spaces based on integralvalue degrees of freedom have the optimal convergence in an energy norm. Following a similar approach, we prove that the interior penalty discontinuous Galerkin schemes based on these IFE functions also have the optimal convergence. However, for the PPG schemes based on midpointvalue degrees of freedom, we prove that they have at least a suboptimal convergence. Numerical experiments are provided to demonstrate features of these IFE methods and compare them with other related numerical schemes.
We extend nonconforming IFE schemes to the planar elasticity interface problem with discontinuous Lam"e parameters. Vectorvalued nonconforming rotated Q1 IFE functions with integralvalue degrees of freedom are unisolvent with appropriate interface jump conditions. More importantly, the Galerkin IFE scheme using these vectorvalued nonconforming rotated Q1 IFE functions are "lockingfree" for nearly incompressible elastic materials.
In the last part of this dissertation, we consider potential applications of IFE methods to time dependent PDEs with moving interfaces. Using IFE functions in the discretization in space enables the applicability of the method of lines. CrankNicolson type fully discrete schemes are also developed as alternative approaches for solving moving interface problems. / Ph. D.

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A Linear Immersed Finite Element Space Defined by Actual Interface Curve on Triangular MeshesGuo, Ruchi 17 April 2017 (has links)
In this thesis, we develop the a new immersed finite element(IFE) space formed by piecewise linear polynomials defined on subelements cut by the actual interface curve for solving elliptic interface problems on interface independent meshes. A group of geometric identities and estimates on interface elements are derived. Based on these geometric identities and estimates, we establish a multipoint Taylor expansion of the true solutions and show the estimates for the second order terms in the expansion. Then, we construct the local IFE spaces by imposing the weak jump conditions and nodal value conditions on the piecewise polynomials. The unisolvence of the IFE shape functions is proven by the invertibility of the wellknown ShermanMorrison system. Furthermore we derive a group of fundamental identities about the IFE shape functions, which show that the two polynomial components in an IFE shape function are highly related. Finally we employ these fundamental identities and the multipoint Taylor expansion to derive the estimates for IFE interpolation errors in L2 and semiH1 norms. / Master of Science

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Bilinear Immersed Finite Elements For Interface ProblemsHe, Xiaoming 02 June 2009 (has links)
In this dissertation we discuss bilinear immersed finite elements (IFE) for solving interface problems. The related research works can be categorized into three aspects: (1) the construction of the bilinear immersed finite element spaces; (2) numerical methods based on these IFE spaces for solving interface problems; and (3) the corresponding error analysis. All of these together form a solid foundation for the bilinear IFEs.
The research on immersed finite elements is motivated by many real world applications, in which a simulation domain is often formed by several materials separated from each other by curves or surfaces while a mesh independent of interface instead of a bodyfitting mesh is preferred. The bilinear IFE spaces are nonconforming finite element spaces and the mesh can be independent of interface. The error estimates for the interpolation of a Sobolev function in a bilinear IFE space indicate that this space has the usual approximation capability expected from bilinear polynomials, which is <i>O</i>(<i>h</i>²) in <i>L</i>² norm and <i>O</i>(<i>h</i>) in <i>H</i>¹ norm. Then the immersed spaces are applied in Galerkin, finite volume element (FVE) and discontinuous Galerkin (DG) methods for solving interface problems. Numerical examples show that these methods based on the bilinear IFE spaces have the same optimal convergence rates as those based on the standard bilinear finite element for solutions with certain smoothness. For the symmetric selective immersed discontinuous Galerkin method based on bilinear IFE, we have established its optimal convergence rate. For the Galerkin method based on bilinear IFE, we have also established its convergence.
One of the important advantages of the discontinuous Galerkin method is its flexibility for both <i>p</i> and <i>h</i> mesh refinement. Because IFEs can use a mesh independent of interface, such as a structured mesh, the combination of a DG method and IFEs allows a flexible adaptive mesh independent of interface to be used for solving interface problems. That is, a mesh independent of interface can be refined wherever needed, such as around the interface and the singular source. We also develop an efficient selective immersed discontinuous Galerkin method. It uses the sophisticated discontinuous Galerkin formulation only around the locations needed, but uses the simpler Galerkin formulation everywhere else. This selective formulation leads to an algebraic system with far less unknowns than the immersed DG method without scarifying the accuracy; hence it is far more efficient than the conventional discontinuous Galerkin formulations. / Ph. D.

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Immersed Finite Elements for a Second Order Elliptic Operator and Their ApplicationsZhuang, Qiao 17 June 2020 (has links)
This dissertation studies immersed finite elements (IFE) for a second order elliptic operator and their applications to interface problems of related partial differential equations.
We start with the immersed finite element methods for the second order elliptic operator with a discontinuous coefficient associated with the elliptic interface problems. We introduce an energy norm stronger than the one used in [111]. Then we derive an estimate for the IFE interpolation error with this energy norm using patches of interface elements. We prove both the continuity and coercivity of the bilinear form in a partially penalized IFE (PPIFE) method. These properties allow us to derive an error bound for the PPIFE solution in the energy norm under the standard piecewise $H^2$ regularity assumption instead of the more stringent $H^3$ regularity used in [111]. As an important consequence, this new estimation further enables us to show the optimal convergence in the $L^2$ norm which could not be done by the analysis presented in [111].
Then we consider applications of IFEs developed for the second order elliptic operator to wave propagation and diffusion interface problems. The first application is for the timeharmonic wave interface problem that involves the Helmholtz equation with a discontinuous coefficient. We design PPIFE and DGIFE schemes including the higher degree IFEs for Helmholtz interface problems. We present an error analysis for the symmetric linear/bilinear PPIFE methods. Under the standard piecewise $H^2$ regularity assumption for the exact solution, following Schatz's arguments, we derive optimal error bounds for the PPIFE solutions in both an energy norm and the usual $L^2$ norm provided that the mesh size is sufficiently small.
{In the second group of applications, we focus on the error analysis for IFE methods developed for solving typical timedependent interface problems associated with the second order elliptic operator with a discontinuous coefficient.} For hyperbolic interface problems, which are typical wave propagation interface problems, we reanalyze the fullydiscrete PPIFE method in [143]. We derive the optimal error bounds for this PPIFE method for both an energy norm and the $L^2$ norm under the standard piecewise $H^2$ regularity assumption in the space variable of the exact solution. Simulations for standing and travelling waves are presented to corroborate the results of the error analysis. For parabolic interface problems, which are typical diffusion interface problems, we reanalyze the PPIFE methods in [113]. We prove that these PPIFE methods have the optimal convergence not only in an energy norm but also in the usual $L^2$ norm under the standard piecewise $H^2$ regularity. / Doctor of Philosophy / This dissertation studies immersed finite elements (IFE) for a second order elliptic operator and their applications to a few types of interface problems.
We start with the immersed finite element methods for the second order elliptic operator with a discontinuous coefficient associated with the elliptic interface problem. We can show that the IFE methods for the elliptic interface problems converge optimally when the exact solution has lower regularity than that in the previous publications.
Then we consider applications of IFEs developed for the second order elliptic operator to wave propagation and diffusion interface problems. For interface problems of the Helmholtz equation which models timeHarmonic wave propagations, we design IFE schemes, including higher degree schemes, and derive error estimates for a lower degree scheme. For interface problems of the second order hyperbolic equation which models time dependent wave propagations, we derive better error estimates for the IFE methods and provides numerical simulations for both the standing and traveling waves. For interface problems of the parabolic equation which models the time dependent diffusion, we also derive better error estimates for the IFE methods.

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Mixed Interface Problems of Thermoelastic PseudoOscillationsJentsch, L., Natroshvili, D., Sigua, I. 30 October 1998 (has links) (PDF)
Threedimensional basic and mixed interface problems of the mathematical
theory of thermoelastic pseudooscillations are considered for piecewise homogeneous
anisotropic bodies. Applying the method of boundary potentials and the theory of
pseudodifferential equations existence and uniqueness theorems of solutions are proved
in the space of regular functions C^(k+ alpha) and in the Besselpotential (H^(s)_(p))
and Besov (B^(s)_(p,q)) spaces. In addition to the classical regularity results
for solutions to the basic interface problems, it is shown that in the mixed interface
problems the displacement vector and the temperature are Hölder continuous with
exponent 0<alpha<1/2.

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Méthodes hybrides d'ordre élevé pour les problèmes d'interface / Hybrid highorder methods for interface problemsChave, Florent 12 November 2018 (has links)
Le but de cette thèse est de développer et d’analyser les méthodes Hybrides d’Ordre Élevé (HHO: Hybrid HighOrder, en anglais) pour des problèmes d’interfaces. Nous nous intéressons à deux types d’interfaces (i) les interfaces diffuses, et (ii) les interfaces traitées comme frontières internes du domaine computationnel. La première moitié de ce manuscrit est consacrée aux interfaces diffuses, et plus précisément aux célèbres équations de Cahn–Hilliard qui modélisent le processus de séparation de phase par lequel les deux composants d’un fluide binaire se séparent pour former des domaines purs en chaque composant. Dans la deuxième moitié, nous considérons des modèles à dimension hybride pour la simulation d’écoulements de Darcy et de transports passifs en milieu poreux fracturé, dans lequel la fracture est considérée comme un hyperplan (d’où le terme hybride) qui traverse le domaine computationnel. / The purpose of this Ph.D. thesis is to design and analyse Hybrid HighOrder (HHO) methods on some interface problems. By interface, we mean (i) diffuse interface, and (ii) interface as an immersed boundary. The first half of this manuscrit is dedicated to diffuse interface, more precisely we consider the so called Cahn–Hilliard problem that models the process of phase separation, by which the two components of a binary fluid spontaneously separate and form domains pure in each component. In the second half, we deal with the interface as an immersed boundary and consider a hybrid dimensional model for the simulation of Darcy flows and passive transport in fractured porous media, in which the fracture is considered as an hyperplane that crosses our domain of interest.

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