DOCSLIB.ORG

An Introduction to the Finite Element Method (FEM) for Differential Equations in 1D

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
An Introduction to the Finite Element Method (FEM) for Differential Equations in 1D An Introduction to the Finite Element Method (FEM) for Differential Equations in 1D Mohammad Asadzadeh June 24, 2015 Contents 1 Introduction 1 1.1 Ordinarydifferentialequations(ODE) . 1 1.2 Partialdifferentialequations(PDE) . 2 1.2.1 Exercises.......................... 8 2 PolynomialApproximationin1d 9 2.1 Overture.............................. 9 2.1.1 Basis function in nonuniform partition . 14 2.2 Variationalformulationfor(IVP) . 17 2.3 Galerkinfiniteelementmethodfor(2.1.1) . 19 2.4 AGalerkinmethodfor(BVP) . 21 2.4.1 Thenonuniformversion . 26 2.5 Exercises.............................. 27 3 Interpolation,NumericalIntegrationin1d 31 3.1 Preliminaries ........................... 31 3.2 Lagrangeinterpolation . 39 3.3 Numericalintegration,Quadraturerules . 41 3.3.1 Composite rules for uniform partitions . 44 3.3.2 Gaussquadraturerule . 48 4 Two-pointboundaryvalueproblems 53 4.1 ADirichletproblem ....................... 53 4.2 Thefiniteelementmethod(FEM). 58 4.3 Errorestimatesintheenergynorm . 59 4.4 FEM for convection–diffusion–absorption BVPs . 65 4.5 Exercises.............................. 72 iii iv CONTENTS 5 ScalarInitialValueProblems 81 5.1 Solutionformulaandstability . 82 5.2 Finitedifferencemethods. 83 5.3 GalerkinfiniteelementmethodsforIVP . 86 5.3.1 ThecontinuousGalerkinmethod . 87 5.3.2 ThediscontinuousGalerkinmethod . 90 5.4 Exercises.............................. 92 6 InitialBoundaryValueProblemsin1d 95 6.1 Heatequationin1d........................ 95 6.1.1 Stabilityestimates . 96 6.1.2 FEMfortheheatequation. .100 6.1.3 Exercises..........................104 6.2 Thewaveequationin1d . .106 6.2.1 WaveequationasasystemofPDEs . 107 6.2.2 The finite element discretization procedure . 108 6.2.3 Exercises..........................111 A Answers to Exercises 115 BAlgorithmsandMATLABCodes 121 Table of Symbols and Indices 135 CONTENTS v Preface and acknowledgments. This text is an elementary approach to finite element method used in numerical solution of differential equations in one space dimension. The purpose is to introduce students to piecewise poly- nomial approximation of solutions using a minimum amount of theory. The presented material in this note should be accessible to students with knowl- edge of calculus of single- and several-variables and linear algebra. The theory is combined with approximation techniques that are easily implemented by Matlab codes presented at the end. During several years, many colleagues have been involved in the design, presentation and correction of these notes. I wish to thank Niklas Eriksson and Bengt Svensson who have read the entire material and made many valu- able suggestions. Niklas has contributed to a better presentation of the text as well as to simplifications and corrections of many key estimates that has substantially improved the quality of this lecture notes. Bengt has made all xfig figures. The final version is further polished by John Bondestam Malm- berg and Tobias Geb¨ack who, in particular, have many useful input in the Matlab codes. vi CONTENTS void Chapter 1 Introduction In this lecture notes we present an introduction to approximate solutions for differential equations. A differential equation is a relation between a function and its derivatives. In case the derivatives that appear in a differential equa- tion are only with respect to one variable, the differential equation is called ordinary. Otherwise it is called a partial differential equation. For example, du u(t)=0, (1.0.1) dt − is an ordinary differetial equation, whereas ∂u ∂2u =0, (1.0.2) ∂t − ∂x2 ∂u ∂2u is a partial differential (PDE) equation. In (1.0.2) ∂t , ∂x2 denote the partial derivatives. Here t denotes the time variable and x is the space variable. We shall only study one space dimentional equations that are either stationary (time-independent) or time dependent. Our focus will be on the following equations: 1.1 Ordinary differential equations (ODE) An example of population dynamic as in (1.0.1) • du λu(t)= f(t), (1.1.1) dt − 1 2 CHAPTER 1. INTRODUCTION where λ is a constant and f is a source function. A stationary (time-independent) heat equation as • d2u = f(x), (1.1.2) −dx2 A stationary convection-diffusion equation • d2u du + = f(x), (1.1.3) −dx2 dx where f(x) is a source function. 1.2 Partial differential equations (PDE) The heat equation • ∂u ∂2u = f(x). (1.2.1) ∂t − ∂x2 The wave equation • ∂2u ∂2u = f(x). (1.2.2) ∂t2 − ∂x2 The time depending convection-diffusion or reaction-diffusion equation • ∂u ∂2u ∂u + = f(x). (1.2.3) ∂t − ∂x2 ∂x Some notation. For convenince we shall use the following notation: ∂u ∂2u ∂u ∂2u u˙ = , u¨ = , u′ = , u′′ = . ∂t ∂t2 ∂x ∂x2 Example 1.1 (Initial Conditions). Consider the simple equation u˙(t)= t. Evidently, u(t) = t2/2, is a solution. But, for any constant C, t2/2+ C is also a solution. In this way we have infintely many solutions (one for each constant C). To determine a unique solution we need to supply the equation with one extra condition. Since the time variable t is always assumed to be t 0, if we know the value of u(t), e.g., at the beginning, i.e., the initial ≥ value e.g., u(0) = 3, then u(t)= t2/2+3 is the unique solution to the initial value problem: u˙(t) = t, u(0) = 3. A differential equation associated with initial conditions is an initial value problem. 1.2. PARTIAL DIFFERENTIAL EQUATIONS (PDE) 3 Example 1.2 (Boundary Conditions). Likewise u(x)= x2/2 is a solu- 2 − tion to u′′(x)=1. But also all u(x) = x /2+ Ax + B are solutions, for all arbitrary− constants A and B. Therefore,− to determine a unique solution u(x) we need to determine some fixed values for A and B, hence we need to supply two conditions. Here if e.g., x belongs to a bounded interval, say, [0, 1], then given the boundary values u(0) = 1 and u(1) = 0, we get from u(x) = x2/2+ Ax + B that B = 1 and A = 1/2. Thus the solution to − − the the initial boundary value problem: u′′(x)=1, u(0) = 1, u(1) = 0 is: u(x) = x2/2 x/2+1. The generale− rule is that one should supply as many conditions− as− the highest ordre of the derivative in each variable. So, for example, for the hear equation u˙ u′′ = 0, to get a unique solution we − nedd to supply one initial condition (there is one time derivative in the equa- tion) and two boundary conditions (there are two derivatives in x), whereas for the wave equation u¨ u′′ = 0 we have to give two conditions in each variable x and t. A differential− equation with supplied boundary conditions is a boundary value problem. Objectives: For f being a simple elementary function (a polynomial, a trigonometric, or exponential type function or a combination of them), the equations (1.1.1)-(1.2.3), associated with suitable initial and boundary condi- tions, have often closed form analytic solutions. But real problems: general two and three dimensional problems, modeled by equations with variable cefficients and in complex geometry, are seldom analytically solvable. In this note our objective is to introduce numerical methods that ap- proximate solutions for differential equations by polynomials. To check the quality (reliability and efficiency) of these numerical methods, we choose to apply them to the equations (1.1.1)-(1.2.3), where we already know their an- alytic solutions. Below we shall give examples of analytic solutions to ODEs: (1.1.1)-(1.1.3). For examples on analytic solutions for the PDEs: (1.2.1)- (1.2.3), we refer to the separation of variables technique introduced in the second part of our course. Example 1.3. Determine the solution to the initial value problem u˙(t) λu(t)=0, u(0) = u , (1.2.4) − 0 assuming that u(t) > 0, for all t, λ =1 and u0 =2. Solution. Since u(t) = 0, for all t, we may divide the equation (1.2.4) by 6 4 CHAPTER 1. INTRODUCTION u˙ (t) u(t) and get u(t) = λ. Relabeling t by s and integrating over (0,t) we get t t t u˙(s) t ds = λ ds = ln u(s) = λ[s]0. (1.2.5) 0 u(s) 0 ⇒ 0 Z Z h i Hence we have u(t) ln u(t) ln u(0) = λt or ln = λt. (1.2.6) − u(0) Thus u(t) = eλt, i.e. u(t)= u eλt. (1.2.7) u(0) 0 t Consequently, with λ = 1 and u0 = 2 we have u(t)=2e . To derive solutions to our examples on a systematic way, we recall the pro- cedure for determining a particular solution up to a second order differential equation with constant coefficiets of the form: u′′(x)+ au′(x)+ bu(x)= f(x). (1.2.8) 1. If f(x) = a polynomial of degree n. Set i) u (x)= a + a x + + a xn, if b =0 p 0 1 ··· n 6 ii) u (x)= x(a + a x + + a xn), if b =0,a =0 p 0 1 ··· n 6 2. If f(x) = (polynom) eσx. Set × σx i) up(x)= z(x)e . This gives a new differential equation for z solved by 1). σx ii) up(x)= Ae , if polynom = constant. This works if σ2 + aσ + b = 0: i.e. σ is not a root to the characteristic equation. 6 3. If f(x)= p cos(ωx)+ q sin(ωx). Set 2 i) up(x) = C cos(ωx)+ D sin(ωx), for ω + aiω + b = 0, i.e., if iω is not a root to the characteristic− equation.6 ii) u (x)= x(C cos(ωx)+D sin(ωx)), if ω2 +aiω +b = 0. p − 1.2. PARTIAL DIFFERENTIAL EQUATIONS (PDE) 5 Example 1.4. Find all solutions to the differential equation u′′(x) u(x) = cos(x).
Load more

Recommended publications
  • Linear Approximation of the Derivative
    Section 3.9 - Linear Approximation and the Derivative Idea: When we zoom in on a smooth function and its tangent line, they are very close to- gether. So for a function that is hard to evaluate, we can use the tangent line to approximate values of the derivative. p 3 Example Usep a tangent line to approximatep 8:1 without a calculator. Let f(x) = 3 x = x1=3. We know f(8) = 3 8 = 2. We'll use the tangent line at x = 8. 1 1 1 1 f 0(x) = x−2=3 so f 0(8) = (8)−2=3 = · 3 3 3 4 Our tangent line passes through (8; 2) and has slope 1=12: 1 y = (x − 8) + 2: 12 To approximate f(8:1) we'll find the value of the tangent line at x = 8:1. 1 y = (8:1 − 8) + 2 12 1 = (:1) + 2 12 1 = + 2 120 241 = 120 p 3 241 So 8:1 ≈ 120 : p How accurate was this approximation? We'll use a calculator now: 3 8:1 ≈ 2:00829885. 241 −5 Taking the difference, 2:00829885 − 120 ≈ −3:4483 × 10 , a pretty good approximation! General Formulas for Linear Approximation The tangent line to f(x) at x = a passes through the point (a; f(a)) and has slope f 0(a) so its equation is y = f 0(a)(x − a) + f(a) The tangent line is the best linear approximation to f(x) near x = a so f(x) ≈ f(a) + f 0(a)(x − a) this is called the local linear approximation of f(x) near x = a.
    [Show full text]
  • Generalized Finite-Difference Schemes
    Generalized Finite-Difference Schemes By Blair Swartz* and Burton Wendroff** Abstract. Finite-difference schemes for initial boundary-value problems for partial differential equations lead to systems of equations which must be solved at each time step. Other methods also lead to systems of equations. We call a method a generalized finite-difference scheme if the matrix of coefficients of the system is sparse. Galerkin's method, using a local basis, provides unconditionally stable, implicit generalized finite-difference schemes for a large class of linear and nonlinear problems. The equations can be generated by computer program. The schemes will, in general, be not more efficient than standard finite-difference schemes when such standard stable schemes exist. We exhibit a generalized finite-difference scheme for Burgers' equation and solve it with a step function for initial data. | 1. Well-Posed Problems and Semibounded Operators. We consider a system of partial differential equations of the following form: (1.1) du/dt = Au + f, where u = (iti(a;, t), ■ ■ -, umix, t)),f = ifxix, t), ■ • -,fmix, t)), and A is a matrix of partial differential operators in x — ixx, • • -, xn), A = Aix,t,D) = JjüiD', D<= id/dXx)h--Yd/dxn)in, aiix, t) = a,-,...i„(a;, t) = matrix . Equation (1.1) is assumed to hold in some n-dimensional region Í2 with boundary An initial condition is imposed in ti, (1.2) uix, 0) = uoix) , xEV. The boundary conditions are that there is a collection (B of operators Bix, D) such that (1.3) Bix,D)u = 0, xEdti,B<E(2>. We assume the operator A satisfies the following condition : There exists a scalar product {,) such that for all sufficiently smooth functions <¡>ix)which satisfy (1.3), (1.4) 2 Re {A4,,<t>) ^ C(<b,<p), 0 < t ^ T , where C is a constant independent of <p.An operator A satisfying (1.4) is called semi- bounded.
    [Show full text]
  • A FETI Method with a Mesh Independent Condition Number for the Iteration Matrix Christine Bernardi, Tomas Chacon-Rebollo, Eliseo Chacòn Vera
    A FETI method with a mesh independent condition number for the iteration matrix Christine Bernardi, Tomas Chacon-Rebollo, Eliseo Chacòn Vera To cite this version: Christine Bernardi, Tomas Chacon-Rebollo, Eliseo Chacòn Vera. A FETI method with a mesh in- dependent condition number for the iteration matrix. Computer Methods in Applied Mechanics and Engineering, Elsevier, 2008, 197 (issues-13-16), pp.1410-1429. 10.1016/j.cma.2007.11.019. hal- 00170169 HAL Id: hal-00170169 https://hal.archives-ouvertes.fr/hal-00170169 Submitted on 6 Sep 2007 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. A FETI method with a mesh independent condition number for the iteration matrix C. Bernardi1, T . Chac´on Rebollo2, E. Chac´on Vera2 August 21, 2007 Abstract. We introduce a framework for FETI methods using ideas from the 1 decomposition via Lagrange multipliers of H0 (Ω) derived by Raviart-Thomas [22] and complemented with the detailed work on polygonal domains devel- oped by Grisvard [17]. We compute the action of the Lagrange multipliers 1/2 using the natural H00 scalar product, therefore no consistency error appears. As a byproduct, we obtain that the condition number for the iteration ma- trix is independent of the mesh size and there is no need for preconditioning.
    [Show full text]
  • Hp-Finite Element Methods for Hyperbolic Problems
    ¢¢¢¢¢¢¢¢¢¢¢ ¢¢¢¢¢¢¢¢¢¢¢ ¢¢¢¢¢¢¢¢¢¢¢ ¢¢¢¢¢¢¢¢¢¢¢ Eidgen¨ossische Ecole polytechnique f´ed´erale de Zurich ¢¢¢¢¢¢¢¢¢¢¢ ¢¢¢¢¢¢¢¢¢¢¢ ¢¢¢¢¢¢¢¢¢¢¢ ¢¢¢¢¢¢¢¢¢¢¢ Technische Hochschule Politecnico federale di Zurigo ¢¢¢¢¢¢¢¢¢¢¢ ¢¢¢¢¢¢¢¢¢¢¢ ¢¢¢¢¢¢¢¢¢¢¢ ¢¢¢¢¢¢¢¢¢¢¢ Z¨urich Swiss Federal Institute of Technology Zurich hp-Finite Element Methods for Hyperbolic Problems E. S¨uli†, P. Houston† and C. Schwab ∗ Research Report No. 99-14 July 1999 Seminar f¨urAngewandte Mathematik Eidgen¨ossische Technische Hochschule CH-8092 Z¨urich Switzerland †Oxford University Computing Laboratory, Wolfson Building, Parks Road, Oxford OX1 3QD, United Kingdom ∗E. S¨uliand P. Houston acknowledge the financial support of the EPSRC (Grant GR/K76221) hp-Finite Element Methods for Hyperbolic Problems E. S¨uli†, P. Houston† and C. Schwab ∗ Seminar f¨urAngewandte Mathematik Eidgen¨ossische Technische Hochschule CH-8092 Z¨urich Switzerland Research Report No. 99-14 July 1999 Abstract This paper is devoted to the a priori and a posteriori error analysis of the hp-version of the discontinuous Galerkin finite element method for partial differential equations of hyperbolic and nearly-hyperbolic character. We consider second-order partial dif- ferential equations with nonnegative characteristic form, a large class of equations which includes convection-dominated diffusion problems, degenerate elliptic equa- tions and second-order problems of mixed elliptic-hyperbolic-parabolic type. An a priori error bound is derived for the method in the so-called DG-norm which is optimal in terms of the mesh size h; the error bound is either 1 degree or 1/2 de- gree below optimal in terms of the polynomial degree p, depending on whether the problem is convection-dominated, or diffusion-dominated, respectively. In the case of a first-order hyperbolic equation the error bound is hp-optimal in the DG-norm.
    [Show full text]
  • An Adaptive Hp-Refinement Strategy with Computable Guaranteed Bound
    An adaptive hp-refinement strategy with computable guaranteed bound on the error reduction factor∗ Patrik Danielyz Alexandre Ernzy Iain Smearsx Martin Vohral´ıkyz April 24, 2018 Abstract We propose a new practical adaptive refinement strategy for hp-finite element approximations of elliptic problems. Following recent theoretical developments in polynomial-degree-robust a posteriori error analysis, we solve two types of discrete local problems on vertex-based patches. The first type involves the solution on each patch of a mixed finite element problem with homogeneous Neumann boundary conditions, which leads to an H(div; Ω)-conforming equilibrated flux. This, in turn, yields a guaranteed upper bound on the error and serves to mark mesh vertices for refinement via D¨orfler’s bulk-chasing criterion. The second type of local problems involves the solution, on patches associated with marked vertices only, of two separate primal finite element problems with homogeneous Dirichlet boundary conditions, which serve to decide between h-, p-, or hp-refinement. Altogether, we show that these ingredients lead to a computable guaranteed bound on the ratio of the errors between successive refinements (error reduction factor). In a series of numerical experiments featuring smooth and singular solutions, we study the performance of the proposed hp-adaptive strategy and observe exponential convergence rates. We also investigate the accuracy of our bound on the reduction factor by evaluating the ratio of the predicted reduction factor relative to the true error reduction, and we find that this ratio is in general quite close to the optimal value of one. Key words: a posteriori error estimate, adaptivity, hp-refinement, finite element method, error reduction, equilibrated flux, residual lifting 1 Introduction Adaptive discretization methods constitute an important tool in computational science and engineering.
    [Show full text]
  • Calculus I - Lecture 15 Linear Approximation & Differentials
    Calculus I - Lecture 15 Linear Approximation & Differentials Lecture Notes: http://www.math.ksu.edu/˜gerald/math220d/ Course Syllabus: http://www.math.ksu.edu/math220/spring-2014/indexs14.html Gerald Hoehn (based on notes by T. Cochran) March 11, 2014 Equation of Tangent Line Recall the equation of the tangent line of a curve y = f (x) at the point x = a. The general equation of the tangent line is y = L (x) := f (a) + f 0(a)(x a). a − That is the point-slope form of a line through the point (a, f (a)) with slope f 0(a). Linear Approximation It follows from the geometric picture as well as the equation f (x) f (a) lim − = f 0(a) x a x a → − f (x) f (a) which means that x−a f 0(a) or − ≈ f (x) f (a) + f 0(a)(x a) = L (x) ≈ − a for x close to a. Thus La(x) is a good approximation of f (x) for x near a. If we write x = a + ∆x and let ∆x be sufficiently small this becomes f (a + ∆x) f (a) f (a)∆x. Writing also − ≈ 0 ∆y = ∆f := f (a + ∆x) f (a) this becomes − ∆y = ∆f f 0(a)∆x ≈ In words: for small ∆x the change ∆y in y if one goes from x to x + ∆x is approximately equal to f 0(a)∆x. Visualization of Linear Approximation Example: a) Find the linear approximation of f (x) = √x at x = 16. b) Use it to approximate √15.9. Solution: a) We have to compute the equation of the tangent line at x = 16.
    [Show full text]
  • Newton-Krylov-BDDC Solvers for Nonlinear Cardiac Mechanics
    Newton-Krylov-BDDC solvers for nonlinear cardiac mechanics Item Type Article Authors Pavarino, L.F.; Scacchi, S.; Zampini, Stefano Citation Newton-Krylov-BDDC solvers for nonlinear cardiac mechanics 2015 Computer Methods in Applied Mechanics and Engineering Eprint version Post-print DOI 10.1016/j.cma.2015.07.009 Publisher Elsevier BV Journal Computer Methods in Applied Mechanics and Engineering Rights NOTICE: this is the author’s version of a work that was accepted for publication in Computer Methods in Applied Mechanics and Engineering. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be reflected in this document. Changes may have been made to this work since it was submitted for publication. A definitive version was subsequently published in Computer Methods in Applied Mechanics and Engineering, 18 July 2015. DOI:10.1016/ j.cma.2015.07.009 Download date 07/10/2021 05:34:35 Link to Item http://hdl.handle.net/10754/561071 Accepted Manuscript Newton-Krylov-BDDC solvers for nonlinear cardiac mechanics L.F. Pavarino, S. Scacchi, S. Zampini PII: S0045-7825(15)00221-2 DOI: http://dx.doi.org/10.1016/j.cma.2015.07.009 Reference: CMA 10661 To appear in: Comput. Methods Appl. Mech. Engrg. Received date: 13 December 2014 Revised date: 3 June 2015 Accepted date: 8 July 2015 Please cite this article as: L.F. Pavarino, S. Scacchi, S. Zampini, Newton-Krylov-BDDC solvers for nonlinear cardiac mechanics, Comput. Methods Appl. Mech. Engrg. (2015), http://dx.doi.org/10.1016/j.cma.2015.07.009 This is a PDF file of an unedited manuscript that has been accepted for publication.
    [Show full text]
  • Finite Difference and Discontinuous Galerkin Methods for Wave Equations
    Digital Comprehensive Summaries of Uppsala Dissertations from the Faculty of Science and Technology 1522 Finite Difference and Discontinuous Galerkin Methods for Wave Equations SIYANG WANG ACTA UNIVERSITATIS UPSALIENSIS ISSN 1651-6214 ISBN 978-91-554-9927-3 UPPSALA urn:nbn:se:uu:diva-320614 2017 Dissertation presented at Uppsala University to be publicly examined in Room 2446, Polacksbacken, Lägerhyddsvägen 2, Uppsala, Tuesday, 13 June 2017 at 10:15 for the degree of Doctor of Philosophy. The examination will be conducted in English. Faculty examiner: Professor Thomas Hagstrom (Department of Mathematics, Southern Methodist University). Abstract Wang, S. 2017. Finite Difference and Discontinuous Galerkin Methods for Wave Equations. Digital Comprehensive Summaries of Uppsala Dissertations from the Faculty of Science and Technology 1522. 53 pp. Uppsala: Acta Universitatis Upsaliensis. ISBN 978-91-554-9927-3. Wave propagation problems can be modeled by partial differential equations. In this thesis, we study wave propagation in fluids and in solids, modeled by the acoustic wave equation and the elastic wave equation, respectively. In real-world applications, waves often propagate in heterogeneous media with complex geometries, which makes it impossible to derive exact solutions to the governing equations. Alternatively, we seek approximated solutions by constructing numerical methods and implementing on modern computers. An efficient numerical method produces accurate approximations at low computational cost. There are many choices of numerical methods for solving partial differential equations. Which method is more efficient than the others depends on the particular problem we consider. In this thesis, we study two numerical methods: the finite difference method and the discontinuous Galerkin method.
    [Show full text]
  • Combining Perturbation Theory and Transformation Electromagnetics for Finite Element Solution of Helmholtz-Type Scattering Probl
    Journal of Computational Physics 274 (2014) 883–897 Contents lists available at ScienceDirect Journal of Computational Physics www.elsevier.com/locate/jcp Combining perturbation theory and transformation electromagnetics for finite element solution of Helmholtz-type scattering problems ∗ Mustafa Kuzuoglu a, Ozlem Ozgun b, a Dept. of Electrical and Electronics Engineering, Middle East Technical University, 06531 Ankara, Turkey b Dept. of Electrical and Electronics Engineering, TED University, 06420 Ankara, Turkey a r t i c l e i n f o a b s t r a c t Article history: A numerical method is proposed for efficient solution of scattering from objects with Received 31 August 2013 weakly perturbed surfaces by combining the perturbation theory, transformation electro- Received in revised form 13 June 2014 magnetics and the finite element method. A transformation medium layer is designed over Accepted 15 June 2014 the smooth surface, and the material parameters of the medium are determined by means Available online 5 July 2014 of a coordinate transformation that maps the smooth surface to the perturbed surface. The Keywords: perturbed fields within the domain are computed by employing the material parameters Perturbation theory and the fields of the smooth surface as source terms in the Helmholtz equation. The main Transformation electromagnetics advantage of the proposed approach is that if repeated solutions are needed (such as in Transformation medium Monte Carlo technique or in optimization problems requiring multiple solutions for a set of Metamaterials perturbed surfaces), computational resources considerably decrease because a single mesh Coordinate transformation is used and the global matrix is formed only once.
    [Show full text]
  • Chapter 3, Lecture 1: Newton's Method 1 Approximate
    Math 484: Nonlinear Programming1 Mikhail Lavrov Chapter 3, Lecture 1: Newton's Method April 15, 2019 University of Illinois at Urbana-Champaign 1 Approximate methods and asumptions The final topic covered in this class is iterative methods for optimization. These are meant to help us find approximate solutions to problems in cases where finding exact solutions would be too hard. There are two things we've taken for granted before which might actually be too hard to do exactly: 1. Evaluating derivatives of a function f (e.g., rf or Hf) at a given point. 2. Solving an equation or system of equations. Before, we've assumed (1) and (2) are both easy. Now we're going to figure out what to do when (2) and possibly (1) is hard. For example: • For a polynomial function of high degree, derivatives are straightforward to compute, but it's impossible to solve equations exactly (even in one variable). • For a function we have no formula for (such as the value function MP(z) from Chapter 5, for instance) we don't have a good way of computing derivatives, and they might not even exist. 2 The classical Newton's method Eventually we'll get to optimization problems. But we'll begin with Newton's method in its basic form: an algorithm for approximately finding zeroes of a function f : R ! R. This is an iterative algorithm: starting with an initial guess x0, it makes a better guess x1, then uses it to make an even better guess x2, and so on. We hope that eventually these approach a solution.
    [Show full text]
  • 2.8 Linear Approximation and Differentials
    196 the derivative 2.8 Linear Approximation and Differentials Newton’s method used tangent lines to “point toward” a root of a function. In this section we examine and use another geometric charac- teristic of tangent lines: If f is differentiable at a, c is close to a and y = L(x) is the line tangent to f (x) at x = a then L(c) is close to f (c). We can use this idea to approximate the values of some commonly used functions and to predict the “error” or uncertainty in a compu- tation if we know the “error” or uncertainty in our original data. At the end of this section, we will define a related concept called the differential of a function. Linear Approximation Because this section uses tangent lines extensively, it is worthwhile to recall how we find the equation of the line tangent to f (x) where x = a: the tangent line goes through the point (a, f (a)) and has slope f 0(a) so, using the point-slope form y − y0 = m(x − x0) for linear equations, we have y − f (a) = f 0(a) · (x − a) ) y = f (a) + f 0(a) · (x − a). If f is differentiable at x = a then an equation of the line L tangent to f at x = a is: L(x) = f (a) + f 0(a) · (x − a) Example 1. Find a formula for L(x), the linear function tangent to the p ( ) = ( ) ( ) ( ) graph of f x p x atp the point 9, 3 . Evaluate L 9.1 and L 8.88 to approximate 9.1 and 8.88.
    [Show full text]
  • A Discontinuous Galerkin Method for Nonlinear Parabolic Equations and Gradient flow Problems with Interaction Potentials
    A discontinuous Galerkin method for nonlinear parabolic equations and gradient flow problems with interaction potentials Zheng Sun1, Jos´eA. Carrillo2 and Chi-Wang Shu3 Abstract We consider a class of time dependent second order partial differential equations governed by a decaying entropy. The solution usually corresponds to a density distribution, hence positivity (non-negativity) is expected. This class of problems covers important cases such as Fokker-Planck type equations and aggregation models, which have been studied intensively in the past decades. In this paper, we design a high order discontinuous Galerkin method for such problems. If the interaction potential is not involved, or the interaction is defined by a smooth kernel, our semi-discrete scheme admits an entropy inequality on the discrete level. Furthermore, by applying the positivity-preserving limiter, our fully discretized scheme produces non-negative solutions for all cases under a time step constraint. Our method also applies to two dimensional problems on Cartesian meshes. Numerical examples are given to confirm the high order accuracy for smooth test cases and to demonstrate the effectiveness for preserving long time asymptotics. Keywords: discontinuous Galerkin method, positivity-preserving, entropy-entropy dissipation relationship, nonlinear parabolic equation, gradient flow 1Division of Applied Mathematics, Brown University, Providence, RI 02912, USA. E-mail: zheng [email protected] 2Department of Mathematics, Imperial College London, London SW7 2AZ, UK. E-mail: car- [email protected]. Research supported by the Royal Society via a Wolfson Research Merit Award and by EPSRC grant number EP/P031587/1. 3Division of Applied Mathematics, Brown University, Providence, RI 02912, USA. E-mail: [email protected].
    [Show full text]