Surface charging is a fundamental interaction process in space plasma engineering. A three-dimensional Immersed Finite Element Particle-In-Cell (IFE-PIC) method is developed to model surface charging involving complex boundary conditions. This method extends the previous IFE-PIC algorithm to explicitly include charge deposition on a dielectric surface for charging calculations. Three simulation studies are carried out using the new algorithm to model current collection and charging in both the orbital motion limited (OML) and space charge limited regime. The first one is a full particle simulation of the charging process of single small sphere and clusters of multiple small spheres in plasma. We find that while single sphere charging agrees well with the predictions of the OML theory, the charging of a sphere in a cluster is significantly, indicating that the often used OML charging model is not an accurate one to model charging in dusty plasma. The second one concerns a secondary electron emission experiment. The simulation includes detailed experimental setup in a vacuum chamber and the results are compared against experimental data. The simulation is used to determine the facility error in experiments. The third one is a full particle simulation of charging on lunar surface. The simulation concerns both flat and non-flat surface, and spacecraft on lunar surface, in the lunar polar region. The surface sees a mesothermal solar wind plasma flow and the emission of photoelectrons and secondary electrons. At a small sun elevation angle, the surface landscape generates a complex plasma flow field and local differential charging on surface. The results will be useful for further study of charging and levitation of lunar dust. / Ph. D.

In this dissertation we prove the superconvergence of the minimal-dissipation local discontinuous Galerkin method for elliptic problems and construct optimal immersed finite element approximations and discontinuous immersed finite element methods for the Stokes interface problem. In the first part we present an error analysis for the minimal dissipation local discontinuous Galerkin method applied to a model elliptic problem on Cartesian meshes when polynomials of degree at most <i>k</i> and an appropriate approximation of the boundary condition are used. This special approximation allows us to achieve <i>k</i> + 1 order of convergence for both the potential and its gradient in the L² norm. Here we improve on existing estimates for the solution gradient by a factor &#8730;h. In the second part we present discontinuous immersed finite element (IFE) methods for the Stokes interface problem on Cartesian meshes that does not require the mesh to be aligned with the interface. As such, we allow unfitted meshes that are cut by the interface. Thus, elements may contain more than one fluid. On these unfitted meshes we construct an immersed Q₁/Q₀ finite element approximation that depends on the location of the interface. We discuss the basic features of the proposed Q₁/Q₀ IFE basis functions such as the unisolvent property. We present several numerical examples to demonstrate that the proposed IFE approximations applied to solve interface Stokes problems maintain the optimal approximation capability of their standard counterpart applied to solve the homogeneous Stokes problem. Similarly, we also show that discontinuous Galerkin IFE solutions of the Stokes interface problem maintain the optimal convergence rates in both L² and broken H¹ norms. Furthermore, we extend our method to solve the axisymmetric Stokes interface problem with a moving interface and test the proposed method by solving several benchmark problems from the literature. / Ph. D.

In this dissertation, we provide a unified framework for analyzing immersed finite element methods in one spatial dimension, and we design a new geometry conforming IFE space in two dimensions with optimal approximation capabilities, alongside with applications to the elliptic interface problem and the hyperbolic interface problem. In the first part, we discuss a general m-th degree IFE space for one dimensional interface problems with many polynomial-like properties, then we develop a general framework for obtaining error estimates for the IFE spaces developed for solving a variety of interface problems, including but not limited to, the elliptic interface problem, the Euler-Bernoulli beam interface problem, the parabolic interface problem, the transport interface problem, and the acoustic interface problem. In the second part, we develop a new m-th degree finite element space based on the differential geometry of the interface to solve interface problems in two spatial dimensions. The proposed IFE space has optimal approximation capabilities, easy to construct, and the IFE functions satisfy the interface conditions exactly. We provide several numerical examples to demonstrate that the IFE space yields optimally converging solutions when applied to the elliptic interface problem and the hyperbolic interface problem with a symmetric interior penalty discontinuous Galerkin formulation. / Doctor of Philosophy / Interface problems appear naturally in many physics and Engineering applications where a physical quantity is considered across materials of different physical properties, such as heat transfer or sound propagation through different materials. Typically, these physical phenomena are modelled by partial differential equations with discontinuous coefficients representing the material properties. The main topics of this dissertation are about the development and analysis of immersed finite element methods for interface problems. The IFE method can use interface independent meshes, and employs approximating functions that capture the features of the solution at the interface. Specifically, we provide a unified framework for analyzing one-dimensional IFE problems, and we design a new framework to construct geometry conforming IFE spaces in two dimensions, with applications to the elliptic interface problem and the hyperbolic interface problem.

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 so-called 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 well-known 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 integral-value 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 midpoint-value degrees of freedom, we prove that they have at least a sub-optimal 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. Vector-valued nonconforming rotated Q1 IFE functions with integral-value degrees of freedom are unisolvent with appropriate interface jump conditions. More importantly, the Galerkin IFE scheme using these vector-valued nonconforming rotated Q1 IFE functions are "locking-free" 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. Crank-Nicolson type fully discrete schemes are also developed as alternative approaches for solving moving interface problems. / Ph. D.

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 specially-built 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 curl-free 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 Petrov-Galerkin method is stable for interface problems with high jumps across the interface. Local time-stepping and parallel algorithms are used to speed up computation. Several realistic interface problems such as ether/glycerol, water/methyl-alcohol and water/air with a circular interface are solved to show the stability and robustness of our methods. / Ph. D.

This thesis develops an immersed finite element (IFE) space for numerical simulations arising from beam design with multiple materials. This IFE space is based upon meshes that can be independent of interface of the materials used to form a beam. Both the forward and inverse problems associated with the beam equation are considered. The order of accuracy of this IFE space is numerically investigated from the point of view of both the interpolation and finite element solution of the interface boundary value problems. Both single and multiple interfaces are considered in our numerical simulation. The results demonstrate that this IFE space has the optimal order of approximation capability. / Master of Science

In this thesis, we develop the a new immersed finite element(IFE) space formed by piecewise linear polynomials defined on sub-elements 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 multi-point 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 well-known Sherman-Morrison 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 multi-point Taylor expansion to derive the estimates for IFE interpolation errors in L2 and semi-H1 norms. / Master of Science / Interface problems occur in many mathematical models in science and engineering that are posed on domains consisting of multiple materials. In general, materials in a modeling domain have different physical or chemical properties; thus, the transmission behaviors across the interface between different materials must be considered. Partial differential equations (PDEs) are often employed in these models and their coefficients are usually discontinuous across the material interface. This leads to the so called interface problems for the involved PDEs whose solutions are usually not smooth across the interface, and this non-smoothness is an obstacle for mathematical analysis and numerical computation. In this thesis, we present a new immersed finite element (IFE) space for efficiently solving a class of interface problems on interface independent meshes. The new IFE space is formed by piecewise linear polynomials defined on sub-elements cut by the actual interface. We present the construction procedure for this IFE space and establish fundamental properties for its shape functions. Furthermore, we prove that the proposed IFE space has the optimal approximation capability.

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 time-harmonic 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 time-dependent 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 fully-discrete 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 time-Harmonic 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.

Speckle noise occurs in a wide range of images due to sampling and digital degradation. Understanding how noise can be present in images have led to multiple denoising techniques. Most of these denoising techniques assume equal noise distribution. When the noise present in the image is not uniform, the resulting denoised image becomes less than the highest standard or quality. For this research, we will be focusing on speckle noise. Unlike Gaussian noise, which affects single pixels on an image, speckle noise affects multiple pixels. Hence it is not possible to remove speckle noise with the traditional gaussian denoising model. We develope a more accurate speckle denoising model and its stable numerical methods. This model is based on the TV minimization and the associated non-linear PDE and Krissian $et$ $al$.'s speckle noise equation model. A realistic and efficient speckle noise equation model was introduced with an edge enhancing feature by adopting a non-convex functional. An effective numerical scheme was introduced and its stability was proved. Also, while working with TV minimization for non-linear PDE and Krissian $et$ $al$ we used a dual approach for faster computation. This work is based on Chambolle's approach for image denoising. The NLM algorithm takes advantage of the high degree of redundancy of any natural image. Also, the NLM algorithm is very accurate since all pixels contribute for denoising at any given pixel. However, due to non-local averaging, one major drawback is computational cost. For this research, we will discuss new denoising techniques based on NLM and total variation for images contaminated by speckle noise. We introduce blockwise and selective denoising methods based on NLM technique and Partial Differential Equations (PDEs) methods for total variation to enhance computational efficiency. Our PDE methods have shown to be very computational efficient and as mentioned before the NLM process is very accurate.

In this dissertation, we explore applications of partial differential equations with discontinuous coefficients. We consider the nonconforming immersed finite element methods (IFE) for modeling and simulating these partial differential equations. A one-dimensional second-order parabolic initial-boundary value problem with discontinuous coefficients is studied. We propose an extension of the immersed finite element method to a high-order immersed finite element method for solving one-dimensional parabolic interface problems. In addition, we introduce a nonconforming immersed finite element method to solve the two-dimensional parabolic problem with a moving interface. In the nonconforming IFE framework, the degrees of freedom are determined by the average integral value over the element edges. The continuity of the nonconforming IFE framework is in the weak sense in comparison the continuity of the conforming IFE framework. Numerical experiments are provided to demonstrate the features and the robustness of these methods. We introduce a class of lowest-order nonconforming immersed finite element methods for solving two-dimensional Stokes interface problem. On triangular meshes, the Crouzeix-Raviart element is used for velocity approximation, and piecewise constant for pressure. On rectangular meshes, the Rannacher-Turek rotated $Q_1$-$Q_0$ finite element is used. We also consider a new mixed immersed finite element method for the Stokes interface problem on an unfitted mesh. The proposed IFE space uses conforming linear elements for one velocity component and nonconforming linear elements for the other component. The new vector-valued IFE functions are constructed to approximate the interface jump conditions. Basic properties including the unisolvency and the partition of unity of these new IFE methods are discussed. Numerical approximations are observed to converge optimally. Lastly, we apply each class of the new immersed finite element methods to solve the unsteady Stokes interface problem. Based on the new IFE spaces, semi-discrete and full-discrete schemes are developed for solving the unsteady Stokes equations with a stationary or a moving interface. A comparison of the degrees of freedom and number of elements are presented for each method. Numerical experiments are provided to demonstrate the features of these methods.