109 research outputs found
ANALYSIS OF ITERATIVE METHODS FOR THE SOLUTION OF BOUNDARY INTEGRAL EQUATIONS WITH APPLICATIONS TO THE HELMHOLTZ PROBLEM
This thesis is concerned with the numerical solution of boundary integral equations
and the numerical analysis of iterative methods. In the first part, we assume
the boundary to be smooth in order to work with compact operators; while in the
second part we investigate the problem arising from allowing piecewise smooth
boundaries. Although in principle most results of the thesis apply to general problems
of reformulating boundary value problems as boundary integral equations
and their subsequent numerical solutions, we consider the Helmholtz equation
arising from acoustic problems as the main model problem.
In Chapter 1, we present the background material of reformulation of Helmhoitz
boundary value problems into boundary integral equations by either the indirect
potential method or the direct method using integral formulae. The problem of
ensuring unique solutions of integral equations for exterior problems is specifically
discussed. In Chapter 2, we discuss the useful numerical techniques for
solving second kind integral equations. In particular, we highlight the superconvergence
properties of iterated projection methods and the important procedure
of Nystrom interpolation.
In Chapter 3, the multigrid type methods as applied to smooth boundary
integral equations are studied. Using the residual correction principle, we are
able to propose some robust iterative variants modifying the existing methods to
seek efficient solutions. In Chapter 4, we concentrate on the conjugate gradient
method and establish its fast convergence as applied to the linear systems arising from general boundary element equations. For boundary integral equalisations on smooth boundaries we have observed, as the underlying mesh sizes decrease,
faster convergence of multigrid type methods and fixed step convergence of the
conjugate gradient method.
In the case of non-smooth integral boundaries, we first derive the singular
forms of the solution of boundary integral solutions for Dirichlet problems and
then discuss the numerical solution in Chapter 5. Iterative methods such as two
grid methods and the conjugate gradient method are successfully implemented
in Chapter 6 to solve the non-smooth integral equations. The study of two
grid methods in a general setting and also much of the results on the conjugate
gradient method are new. Chapters 3, 4 and 5 are partially based on publications
[4], [5] and [35] respectively.Department of Mathematics and Statistics,
Polytechnic South Wes
Review of Summation-by-parts schemes for initial-boundary-value problems
High-order finite difference methods are efficient, easy to program, scales
well in multiple dimensions and can be modified locally for various reasons
(such as shock treatment for example). The main drawback have been the
complicated and sometimes even mysterious stability treatment at boundaries and
interfaces required for a stable scheme. The research on summation-by-parts
operators and weak boundary conditions during the last 20 years have removed
this drawback and now reached a mature state. It is now possible to construct
stable and high order accurate multi-block finite difference schemes in a
systematic building-block-like manner. In this paper we will review this
development, point out the main contributions and speculate about the next
lines of research in this area
Excitations and spectra from equilibrium real-time Green's functions
The real-time contour formalism for Green's functions provides time-dependent
information of quantum many-body systems. In practice, the long-time simulation
of systems with a wide range of energy scales is challenging due to both the
storage requirements of the discretized Green's function and the computational
cost of solving the Dyson equation. In this manuscript, we apply a real-time
discretization based on a piece-wise high-order orthogonal-polynomial expansion
to address these issues. We present a superconvergent algorithm for solving the
real-time equilibrium Dyson equation using the Legendre spectral method and the
recursive algorithm for Legendre convolution. We show that the compact high
order discretization in combination with our Dyson solver enables long-time
simulations using far fewer discretization points than needed in conventional
multistep methods. As a proof of concept, we compute the molecular spectral
functions of H, LiH, He and CHO using self-consistent
second-order perturbation theory and compare the results with standard quantum
chemistry methods as well as the auxiliary second-order Green's function
perturbation theory method
Discretizations and Solvers for Coupling Stokes-Darcy Flows With Transport
This thesis studies a mathematical model, in which Stokes-Darcy flow system is coupled with a transport equation. The objective is to develop stable and convergent numerical schemes that could be used in environmental applications. Special attention is given to discretization methods that conserve mass locally. First, we present a global saddle point problem approach, which employs the discontinuous Galerkin method to discretize the Stokes equations and the mimetic finite difference method to discretize the Darcy equation. We show how the numerical scheme can be formulated on general polygonal (polyhedral in three dimensions) meshes if suitable operators mapping from degrees of freedom to functional spaces are constructed. The scheme is analyzed and error estimates are derived. A hybridization technique is used to solve the system effectively. We ran several numerical experiments to verify the theoretical convergence rates and depending on the mesh type we observed superconvergence of the computed solution in the Darcy region.Another approach that we use to deal with the flow equations is based on non-overlapping domain decomposition. Domain decomposition enables us to solve the coupled Stokes-Darcy flow problem in parallel by partitioning the computational domain into subdomains, upon which families of coupled local problems of lower complexity are formulated. The coupling of the subdomain problems is removed through an iterative procedure. We investigate the properties of this method and derive estimates for the condition number of the associated algebraic system. Results from computer tests supporting the convergence analysis of the method are provided. To discretize the transport equation we use the local discontinuous Galerkin (LDG) method, which can be thought as a discontinuous mixed finite element method, since it approximates both the concentration and the diffusive flux. We develop stability and convergence analysis for the concentration and the diffusive flux in the transport equation. The numerical error is a combination of the LDG discretization error and the error from the discretization of the Stokes-Darcy velocity. Several examples verifying the theory and illustrating the capabilities of the method are presented
Hägusad teist liiki integraalvõrrandid
Käesolevas doktoritöös on uuritud hägusaid teist liiki integraalvõrrandeid. Need võrrandid sisaldavad hägusaid funktsioone, s.t. funktsioone, mille väärtused on hägusad arvud.
Me tõestasime tulemuse sileda tuumaga hägusate Volterra integraalvõrrandite lahendite sileduse kohta. Kui integraalvõrrandi tuum muudab märki, siis integraalvõrrandi lahend pole üldiselt sile. Nende võrrandite lahendamiseks me vaatlesime kollokatsioonimeetodit tükiti lineaarsete ja tükiti konstantsete funktsioonide ruumis. Kasutades lahendi sileduse tulemusi tõestasime meetodite koonduvuskiiruse.
Me vaatlesime ka nõrgalt singulaarse tuumaga hägusaid Volterra integraalvõrrandeid. Uurisime lahendi olemasolu, ühesust, siledust ja hägusust. Ülesande ligikaudseks lahendamiseks kasutasime kollokatsioonimeetodit tükiti polünoomide ruumis. Tõestasime meetodite koonduvuskiiruse ning uurisime lähislahendi hägusust. Nii analüüs kui ka numbrilised eksperimendid näitavad, et gradueeritud võrke kasutades saame parema koonduvuskiiruse kui ühtlase võrgu korral.
Teist liiki hägusate Fredholmi integraalvõrrandite lahendamiseks pakkusime uue lahendusmeetodi, mis põhineb kõigi võrrandis esinevate funktsioonide lähendamisel Tšebõšovi polünoomidega. Uurisime nii täpse kui ka ligikaudse lahendi olemasolu ja ühesust. Tõestasime meetodi koonduvuse ja lähislahendi hägususe.In this thesis we investigated fuzzy integral equations of the second kind. These equations contain fuzzy functions, i.e. functions whose values are fuzzy numbers.
We proved a regularity result for solution of fuzzy Volterra integral equations with smooth kernels. If the kernel changes sign, then the solution is not smooth in general. We proposed collocation method with triangular and rectangular basis functions for solving these equations. Using the regularity result we estimated the order of convergence of these methods.
We also investigated fuzzy Volterra integral equations with weakly singular kernels. The existence, regularity and the fuzziness of the exact solution is studied. Collocation methods on discontinuous piecewise polynomial spaces are proposed. A convergence analysis is given. The fuzziness of the approximate solution is investigated. Both the analysis and numerical methods show that graded mesh is better than uniform mesh for these problems.
We proposed a new numerical method for solving fuzzy Fredholm integral equations of the second kind. This method is based on approximation of all functions involved by Chebyshev polynomials. We analyzed the existence and uniqueness of both exact and approximate fuzzy solutions. We proved the convergence and fuzziness of the approximate solution.https://www.ester.ee/record=b539569
On the numerical solution of Volterra integral equations on equispaced nodes
In the present paper, a Nystrom-type method for second kind Volterra integral
equations is introduced and studied. The method makes use of generalized
Bernstein polynomials, defined for continuous functions and based on equally
spaced points. Stability and convergence are studied in the space of continuous
functions, and some numerical tests illustrate the performance of the proposed
approach
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