t x 0.25

Size: px
Start display at page:

Download "t x 0.25"

Transcription

1 Journal of ELECTRICAL ENGINEERING, VOL. 52, NO. /s, 2, 48{52 COMPARISON OF BROYDEN AND NEWTON METHODS FOR SOLVING NONLINEAR PARABOLIC EQUATIONS Ivan Cimrak If the time discretization of a nonlinear parabolic dierential equation is realized by Rothe's method then the problem of solving degenerate elliptic equations occurs. After performing a space discretization on each time level the problem can be transformed into solving large systems of nonlinear equations. In this paper Broyden's method and Newton-Kantorovich method are used for solving such systems. The numerical implementations of both methods are compared from the point of view of time eectiveness. Several numerical experiments comparing the Broyden and Newton-Kantorovich methods are studied. K e y w o r d s: Degenerate parabolic equation, slow diusion, quasi-newton method, Broyden method, Newton- Kantorovich method, Barenblatt solution. 2 Mathematics Subject Classication: 35K65, 65P5, 65M6 INTRODUCTION The purpose of this paper is to compare computational results of two methods for solving nonlinear degenerate parabolic equations. Computational eciency of Newton's method is compared with the Broyden quasi- Newton method. These methods are used for solving a degenerate nonlinear elliptic problem in each time step level. We will be mainly concerned with the following nonlinear parabolic equation: t u (u) = ; in (; T ) ; () where is a smoothly bounded domain in R N. A solution to () is subject to the Dirichlet boundary condition and initial condition u = ; (2) u(x; ) = u (x) : (3) The initial-boundary value problem (){(3) will be denoted by (P ) hereafter. We assume that the function satises the assumption (A) is C smooth function such that. Notice that in the case () = the problem (P ) is a degenerated parabolic equation. In general, a solution to (P ) need not be necessarily smooth and that is why we have to deal with weak solutions only. Denition. A function u is called a weak solution to the problem (P ) i (i) u 2 L 2 (I; H ()) \ L (I ); (ii) u satises the integral identity Z T Z (u(x; t) u (x))v t (x; t) r(u(x; t))rv(x; t) dx dt = ; for all v 2 L 2 (I; H ()) such that v t 2 L (I ) and v(; T ) =. It is worth to note that the problem (P ) has at most one weak solution (see [9]). In this paper we investigate numerical aspects of Broyden's and Newton-Kantorovich methods. We implement both methods and compare their eciency and computational time needed for numerical results of the same precision. In [7] a proof of the existence of a solution as well as convergence of Rothe's method used for solving parabolic equations of the form () has been shown. In [6] authors suggested a linear approximation scheme for solving the problem ( P ) and they proved the convergence of approximative solutions to an exact solution. For the numerical implementation of this scheme we refer to [8]. The comparison of this scheme and Broyden's method has been studied in [2]. Recall that the Broyden method as a part of quasi-newton methods is often used in optimization problems like, e.g., minimization of nonlinear functionals in nite dimensional spaces. In the context of nonlinear parabolic equations, the Broyden method has been successfully applied as a tool for minimizing penalty functionals in inverse problems. We refer to recent papers by Soria and Pegon [] and Yu [] for applications of Broyden's method used for solving nonlinear systems arising from optimization problems for nonlinear parabolic equations. However, inspection of the literature shows that this method has been rarely used for solving Department of Mathematical Analysis, Ghent University, Galglaan 2, 9 Ghent, Belgium, cimocage.rug.ac.be ISSN c 2 FEI STU

2 Journal of ELECTRICAL ENGINEERING VOL. 52, NO. /s 2 49 nonlinear systems of equations in innite dimension such as nonlinear elliptic problems. On the other hand, the Newton method and the so-called Jager Kacur method attracted a lot of attention with respect to numerical solution of degenerate parabolic equations and systems. In [4] Fuhrmann investigated a numerical solution to nonlinear parabolic equations like, e.g., Richard's equation or the equation describing the motion of a viscous compressible uid in a porous medium. Special attention was paid to problems with coecients varying in space. In his approach the solution scheme is based on an implicit time discretization combined with Newton's method. For the solution of linear problems, iterative methods are used. The author compares in [3] the Newton method with linear approximation scheme of Jager and Kacur introduced in [6]. TIME DISCRETIZATION As usual, time discretization of a nonlinear parabolic equation () is done by using standard Rothe's step functions. The time interval I = h; T i will be divided into n subintervals ht i ; t i i such that jt i t i j = = n. The time derivative u t is approximated by the backward Euler approximation ui ui. For each xed n we therefore have to solve n elliptic problems of the form u i u i (u i ) = ; (4) u i = : (5) known when computing the new time level v i. If we denote F (v) := v v i f (v) (6) then our aim is to nd a solution v to the following nonlinear equation: F (v) = ; F : R K! R K : (7) 2 SOLUTION IN TIME LEVEL It is dicult to nd an exact solution to the problem (7) in general. There are several numerical methods used in order to nd approximation of the exact solution. We will be concerned with iterative schemes of Newton and quasi-newton types. Hereafter, we will denote all inner iterations of these schemes by an upper index. We will seek for a solution v to (7) as a limit v = lim k! v k, where fv k g k= is a suitable sequence of approximations of v constructed via either Newton's or quasi-newton's (Broyden's) method. The Newton-Kantorovich method This method is commonly used in solving nonlinear problems of the form (7). The so-called Newton's iterations are dened by v k+ = v k [F (v k )] F (v k ) : (8) By solving the above elliptic problems we obtain functions u ; u ; : : : ; u n. Using these functions one can construct Rothe's step function u n : u n (x; t) = u i (x) if t 2 ht i ; t i i : It was proved in [7] that the sequence fu n g converges to a weak solution to the problem (P ) in the norm of the space L 2 (I; H ). Now the main problem is how to nd functions u i. One can discretize the elliptic problem by using several methods, e.g., Finite Elements Method, Finite Dierences Method and others. In any case, the space of functions in which we look for u i is approximated by nite dimensional spaces equivalent to R K. After time and space discretization is performed, functions u i are approximated by vectors v i = (vi ; v2 i ; : : : ; vk i ) 2 RK and problem (P ) is reduced to the following problem v i v i f (v i ) = ; (R) with respect to the approximation of the Dirichlet boundary condition. The operator f approximates the Laplace operator. Of course, it depends on the way how the space discretization has been performed. It is important to emphasize that the vector v i is already We refer to a book [5] by Hutson for a proof of local quadratic convergence of Newton's iterations to a solution of equation (7). Let us recall that assumptions needed for the proof of local quadratic convergence require that the norm kf (v )k is suciently small. This can be guaranteed by taking v = v i and assuming that the time discretization step > is small enough. Broyden's method This method is based on the so-called Broyden's update formula for quasi-newton iterations: v k+ = v k B k F (vk ) : (9) In the N-K method matrices we have B k = F (v k ). In Broyden's method, B k, k = ; 2; : : :, represent only approximations of the Jacobi matrix F (v k ). If we denote s k = v k+ v k ; y k = F (v k+ ) F (v k ) ; then the Broyden update for a new approximation of the Jacobi matrix B k+ is given by B k+ = B k + y k B k s k ks k k 2 s k > :

3 5 I. Cimrak: COMPARISON OF BROYDEN AND NEWTON METHODS FOR SOLVING NONLINEAR PARABOLIC EQUATIONS t x.25.5 Fig.. The exact Barenblatt solution.8 Table. Computational time in the D problem for the spatial grid size h = :35. N-K method Broyden's method :5 :66 : :25 :28 :225 :6 :89 :53 :25 :25 :745 Table 2. Computational time in the D problem for the time step = :25. h N-K method Broyden's method :7 :68 :349 :466 :98 :78 :35 :28 :225 :28 :57 2:9 :233 :87 2:826 :2 :26 3:98 results based on this method. Otherwise, we have to pay for this precision with more operations needed to compute new iterates of the N-K method compared to relatively cheap implementation requirements of the Broyden method. We have considered a simple example of a diusion equation with a known exact solution. Let us consider the following equation t u u m = ; x 2 R N ; t 2 (; T i ; () where m >. This equation can be used in modeling slow diusion problems. Denote z(x; t; a; ) = " (t + ) k a 2 Cjxj2 (t + ) 2k N # m N where [] + = max[; ], k =, C = k(m ) N(m )+2 2Nm and a; > are positive constants. Taking u(x; t) = z(x; t; a; ) one can easily verify that such a function u is a solution to () with the initial condition u(x; ) = z(x; ; a; ). This explicit solution is referred to as the Barenblatt solution to (). The support fx 2 R N j u(x; t) > g of the solution is a bounded subset of R N for every t. (See Fig..) We have tested this example in one and in two spatial dimensions. We focused on the eectiveness of the two methods. In every example we set parameters such as the number of N-K or Broyden's iterations to gain the same precision of numerical solution. Then we measured the computational time needed for both methods. + ; Thanks to Sherman-Morrison formula (see Allgower and Georg []) we can directly compute B k+ if B k is known. That is the main advantage in comparison to solving systems using N-K method. No systems of linear equations with large matrices have to be solved. Such a formula can be derived by the following nice geometric motivation discussed in more detail in []. In this book one can also nd a proof of a local superlinear convergence of Broyden's iterates to the root of (7). The assumptions needed for the proof of a local superlinear convergence require closeness of the initial iterate v and the root v. Again, this requirement can be guaranteed by taking v = v i and assuming <. 3 COMPARISON OF BOTH METHODS In this section we compare the results of implementation of both the Newton-Kantorovich and Broyden methods for solving one time step level problems in time discretization of the degenerate parabolic problem. Since theoretical results indicate a better rate of convergence for the N-K method, we expected more precise numerical One spatial dimension The Laplace operator xx was approximated by nite dierences dened on a uniform spatial mesh with the grid size equals to h. Such an approximation of xx yields a simple three diagonal matrix F from (8). Therefore we can use a fast solver for a linear system of equations with a three diagonal matrix. The time needed to compute numerical solution is apparently shorter for N-K scheme. (See Tables and 2). By using Broyden's scheme we need to compute several scalar products in each time step and this is why, in one spatial dimension, Broyden's method turned to be less eective than the N-K method. Since the support of the Barenblatt solution is bounded, for any time t > we can take suciently large and we can assume zero Dirichlet boundary conditions. We performed experiments in two ways. First we investigated the dependence of the computational TIME for a xed grid size h > and varying time steps. Next we have xed x and we varied the grid size h >. In order to compare precisions of the two methods we used discrete approximations of continuous norms of the Banach spaces L 2 (), L and W 2 () spaces. Let

4 Journal of ELECTRICAL ENGINEERING VOL. 52, NO. /s 2 5 In Tab. 2 the tolerances were set as follows: : < 3 ku NK u g k L 2() < 4:689 ; Fig. 2. The initial prole Table 3. Computational time in the 2D problem for the spatial grid size h = :6. N-K method Broyden's method :5 3:89 :55 :25 7:97 2:3 :6 9:8 3:45 :25 3:372 4:6 Table 4. Computational time in the 2D problem for the time step = :5. h N-K method Broyden's method :6 3:89 :55 :52 4:433 :278 :45 5:54 :47 :39 5:773 :566 :33 6:59 :78 :28 7:32 :844 us dene the following discrete norms of a vector u = (u ; u 2 ; : : : ; u K ): kuk Lp = n kuk L = max K i= nx i= ju i j p p ; () (ju i j) ; (2) :22524 < 3 ku B u gk L2() < 4:5984 ; :388 < 2 ku NK u gk L() < 2:978 ; :378 < 2 ku B u g k L() < 2:967 : Two spatial dimensions The Laplace operator xx + yy was approximated by nite dierences dened on a uniform spatial mesh with a grid size h > in each direction. Such an approximation leads to a more complicated matrix F (see (8)). The matrix is only block diagonal and it was not possible to use a fast solver for this system. When the solver for full matrices is used, the computational time in 2-D for N-K method is about 2 times longer than for Broyden's method. Then, we used the standard package meschach for operations with sparse matrices. Some problems with indefinite matrices occur and the software had to be modied. The computational time needed to compute a numerical solution is rapidly growing for N-K if we enlarge spatial discretization (see Tabs. 3 and 4). Broyden's scheme requires less operations and it seems to be more ecient compared with N-K scheme in 2D. While solving problems in 3D, there are extremely large matrices and using of Broyden's scheme could be very eective. We used discrete L p norms from (2) and () to guarantee the same precision of the methods again. In the case which is shown in Tab. 3 the controlled difference between numerical solutions u NK ; u B and exact solution u g at grid points was estimated by 2:4362 < 2 ku NK u gk L2() < 2:443 ; 2:4344 < 2 ku B u g k L 2() < 2:4378 ; 9:3359 < 2 ku NK u g k L() < 9:3377 ; 9:339 < 2 ku B u gk L() < 9:337 : For the comparison of computational times shown in Tab., as a stopping criterion we assumed the following tolerances for various norms of dierences of numerical solutions u NK ; u B and the exact solution u g : 2:2866 < 3 ku NK u g k L 2() < 2: ; 2:568 < 3 ku B u g k L 2() < 2:986 ; :2984 < 2 ku NK u gk L() < :3448 ; :34 < 2 ku B u g k L() < :3433 : In the last case (see Tab. 4) we obtained the following estimates: :63896 < 2 ku NK u g k L 2() < :93294 ; :63697 < 2 ku B u g k L 2() < :9399 ; 7:3388 < 2 ku NK u gk L() < 8:86 ; 7:336 < 2 ku B u g k L() < 8:7947 :

5 52 I. Cimrak: COMPARISON OF BROYDEN AND NEWTON METHODS FOR SOLVING NONLINEAR PARABOLIC EQUATIONS t= s t=.33 s t=.66 s t=. s t=.33 s t=.66 s Fig. 3. The time evolution of the interface 4 OTHER NUMERICAL EXPERIMENTS We solve one more equation with unknown exact solution (taken from [8]). Let us consider a parabolic equation with homogeneous Dirichlet boundary conditions 2 t u x 2 (um )+ 2 y 2 (um ) +Cu p = ; x 2 ; t 2 (; T ); where = ( L; L) ( L; L) and p; C are positive constants. We consider an initial prole u(x; ) = u (x) (see Fig. 2) where u (x) = 8 < : ; i h x = ; y = ; (x2 +y 2 ) 2 ; xy 6= : (x 6 +y 6 ) 2 The parameters were chosen: m = 2 ; p = :5 ; c = 5 ; = :3 ; T = :2 ; L = :5 ; h = 4 : The evolution of the interface sptfu(; t)g is depicted in Fig. 3. Acknowledgment + [3] FUHRMANN, J. : Numerical Solution Schemes For Nonlinear Diusion Problems Based on Newton's Method, Proceedings of contributed papers and posters, ALGORITMY'97, Conference on Scientic Computing, West Tatra Mountains, September 2-5 (997) pp. 32{4. [4] FUHRMANN, J. : On Numerical Solution Methods for Nonlinear Parabolic Problems, Vieweg. Notes Numer. Fluid Mech. 59 (997), 7{8. [5] HUTSON, V. PYM, J. S. : Applications of Functional Analysis and Operator Theory, Academic Press, London, 98. [6] J AGER, W. KACUR, J. : Solution of Porous Medium Type Systems by Linear Approximation Schemes, Numer. Math. 6 (99), 47{427. [7] KACUR, J. : Method of Rothe in Evolution Equations, BSB Teubner Verlag, Leipzig, 985. [8] MIKULA, K. : Numerical Solution of Nonlinear Diusion with Finite Extinction Phenomenon, Acta Math. Univ. Comenianae 64 No. 2 (995), 73{84. [9] OTTO, F. : L Contraction and Uniqueness for Quasilinear Elliptic-Parabolic Equations, Journal of Dierential Equations 3 (996), 2{38. [] SORIA, A. PEGON, P. : Quasi-Newton Iterative Strategies Applied to the Heat Diusion Equation, Int. J. Numer. Methods Eng. 4 (99), 66{677. [] YU, W. : A Quasi-Newton Method in Innite-Dimensional Spaces and its Application for Solving a Parabolic Inverse Problem, J. Comput. Math. 6 No. 4 (998), 35{38. Received 29 May 2 Revised October 2 The author is greatly indebted to D. Sevcovic for his constructive remarks. References [] ALLGOWER, E. L. GEORG, K. : Numerical Continuation Methods, Springer Verlag, Berlin, 99. [2] CIMR AK, I. SEVCOVIC, D. : Application of Broyden's Method for Solving Nonlinear Degenerate Parabolic Equations, submitted. Ivan Cimrak (Mgr) is a graduate student in mathematics at the Department of Mathematical Analysis of the Faculty of Engeneering, University of Ghent, Belgium. His supervisor is prof. Roger Van Keer. His research interests include electromagnetism, magnetic hysteresis, FEM. As a high school student he parcitipated in International Mathematical Olympiads in Toronto (995) and in Bombay (996) and he won bronze medals. In 2 he nished study of Numerical Analysis at the Faculty of Mathematics, Physics and Informatics, Comenius University, Bratislava, Slovakia.

A MULTIGRID ALGORITHM FOR. Richard E. Ewing and Jian Shen. Institute for Scientic Computation. Texas A&M University. College Station, Texas SUMMARY

A MULTIGRID ALGORITHM FOR. Richard E. Ewing and Jian Shen. Institute for Scientic Computation. Texas A&M University. College Station, Texas SUMMARY A MULTIGRID ALGORITHM FOR THE CELL-CENTERED FINITE DIFFERENCE SCHEME Richard E. Ewing and Jian Shen Institute for Scientic Computation Texas A&M University College Station, Texas SUMMARY In this article,

More information

Mathematics Research Report No. MRR 003{96, HIGH RESOLUTION POTENTIAL FLOW METHODS IN OIL EXPLORATION Stephen Roberts 1 and Stephan Matthai 2 3rd Febr

Mathematics Research Report No. MRR 003{96, HIGH RESOLUTION POTENTIAL FLOW METHODS IN OIL EXPLORATION Stephen Roberts 1 and Stephan Matthai 2 3rd Febr HIGH RESOLUTION POTENTIAL FLOW METHODS IN OIL EXPLORATION Stephen Roberts and Stephan Matthai Mathematics Research Report No. MRR 003{96, Mathematics Research Report No. MRR 003{96, HIGH RESOLUTION POTENTIAL

More information

Spurious Chaotic Solutions of Dierential. Equations. Sigitas Keras. September Department of Applied Mathematics and Theoretical Physics

Spurious Chaotic Solutions of Dierential. Equations. Sigitas Keras. September Department of Applied Mathematics and Theoretical Physics UNIVERSITY OF CAMBRIDGE Numerical Analysis Reports Spurious Chaotic Solutions of Dierential Equations Sigitas Keras DAMTP 994/NA6 September 994 Department of Applied Mathematics and Theoretical Physics

More information

Applied Mathematics 505b January 22, Today Denitions, survey of applications. Denition A PDE is an equation of the form F x 1 ;x 2 ::::;x n ;~u

Applied Mathematics 505b January 22, Today Denitions, survey of applications. Denition A PDE is an equation of the form F x 1 ;x 2 ::::;x n ;~u Applied Mathematics 505b January 22, 1998 1 Applied Mathematics 505b Partial Dierential Equations January 22, 1998 Text: Sobolev, Partial Dierentail Equations of Mathematical Physics available at bookstore

More information

Solution of the Two-Dimensional Steady State Heat Conduction using the Finite Volume Method

Solution of the Two-Dimensional Steady State Heat Conduction using the Finite Volume Method Ninth International Conference on Computational Fluid Dynamics (ICCFD9), Istanbul, Turkey, July 11-15, 2016 ICCFD9-0113 Solution of the Two-Dimensional Steady State Heat Conduction using the Finite Volume

More information

On the solution of incompressible two-phase ow by a p-version discontinuous Galerkin method

On the solution of incompressible two-phase ow by a p-version discontinuous Galerkin method COMMUNICATIONS IN NUMERICAL METHODS IN ENGINEERING Commun. Numer. Meth. Engng 2006; 22:741 751 Published online 13 December 2005 in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/cnm.846

More information

Novel determination of dierential-equation solutions: universal approximation method

Novel determination of dierential-equation solutions: universal approximation method Journal of Computational and Applied Mathematics 146 (2002) 443 457 www.elsevier.com/locate/cam Novel determination of dierential-equation solutions: universal approximation method Thananchai Leephakpreeda

More information

2 The second case, in which Problem (P 1 ) reduces to the \one-phase" problem (P 2 ) 8 >< >: u t = u xx + uu x t > 0, x < (t) ; u((t); t) = q t > 0 ;

2 The second case, in which Problem (P 1 ) reduces to the \one-phase problem (P 2 ) 8 >< >: u t = u xx + uu x t > 0, x < (t) ; u((t); t) = q t > 0 ; 1 ON A FREE BOUNDARY PROBLEM ARISING IN DETONATION THEORY: CONVERGENCE TO TRAVELLING WAVES 1. INTRODUCTION. by M.Bertsch Dipartimento di Matematica Universita di Torino Via Principe Amedeo 8 10123 Torino,

More information

/00 $ $.25 per page

/00 $ $.25 per page Contemporary Mathematics Volume 00, 0000 Domain Decomposition For Linear And Nonlinear Elliptic Problems Via Function Or Space Decomposition UE-CHENG TAI Abstract. In this article, we use a function decomposition

More information

CONVERGENCE OF GAUGE METHOD FOR INCOMPRESSIBLE FLOW CHENG WANG AND JIAN-GUO LIU

CONVERGENCE OF GAUGE METHOD FOR INCOMPRESSIBLE FLOW CHENG WANG AND JIAN-GUO LIU MATHEMATICS OF COMPUTATION Volume 69, Number 232, Pages 135{1407 S 0025-571(00)0124-5 Article electronically published on March 24, 2000 CONVERGENCE OF GAUGE METHOD FOR INCOMPRESSIBLE FLOW CHENG WANG AND

More information

Lecture 9 Approximations of Laplace s Equation, Finite Element Method. Mathématiques appliquées (MATH0504-1) B. Dewals, C.

Lecture 9 Approximations of Laplace s Equation, Finite Element Method. Mathématiques appliquées (MATH0504-1) B. Dewals, C. Lecture 9 Approximations of Laplace s Equation, Finite Element Method Mathématiques appliquées (MATH54-1) B. Dewals, C. Geuzaine V1.2 23/11/218 1 Learning objectives of this lecture Apply the finite difference

More information

The method of lines (MOL) for the diffusion equation

The method of lines (MOL) for the diffusion equation Chapter 1 The method of lines (MOL) for the diffusion equation The method of lines refers to an approximation of one or more partial differential equations with ordinary differential equations in just

More information

The Newton-ADI Method for Large-Scale Algebraic Riccati Equations. Peter Benner.

The Newton-ADI Method for Large-Scale Algebraic Riccati Equations. Peter Benner. The Newton-ADI Method for Large-Scale Algebraic Riccati Equations Mathematik in Industrie und Technik Fakultät für Mathematik Peter Benner benner@mathematik.tu-chemnitz.de Sonderforschungsbereich 393 S

More information

Numerical methods for the Navier- Stokes equations

Numerical methods for the Navier- Stokes equations Numerical methods for the Navier- Stokes equations Hans Petter Langtangen 1,2 1 Center for Biomedical Computing, Simula Research Laboratory 2 Department of Informatics, University of Oslo Dec 6, 2012 Note:

More information

n i,j+1/2 q i,j * qi+1,j * S i+1/2,j

n i,j+1/2 q i,j * qi+1,j * S i+1/2,j Helsinki University of Technology CFD-group/ The Laboratory of Applied Thermodynamics MEMO No CFD/TERMO-5-97 DATE: December 9,997 TITLE A comparison of complete vs. simplied viscous terms in boundary layer

More information

Numerical Methods for the Landau-Lifshitz-Gilbert Equation

Numerical Methods for the Landau-Lifshitz-Gilbert Equation Numerical Methods for the Landau-Lifshitz-Gilbert Equation L ubomír Baňas Department of Mathematical Analysis, Ghent University, 9000 Gent, Belgium lubo@cage.ugent.be http://cage.ugent.be/~lubo Abstract.

More information

Linear Algebra (part 1) : Vector Spaces (by Evan Dummit, 2017, v. 1.07) 1.1 The Formal Denition of a Vector Space

Linear Algebra (part 1) : Vector Spaces (by Evan Dummit, 2017, v. 1.07) 1.1 The Formal Denition of a Vector Space Linear Algebra (part 1) : Vector Spaces (by Evan Dummit, 2017, v. 1.07) Contents 1 Vector Spaces 1 1.1 The Formal Denition of a Vector Space.................................. 1 1.2 Subspaces...................................................

More information

Tutorial 2. Introduction to numerical schemes

Tutorial 2. Introduction to numerical schemes 236861 Numerical Geometry of Images Tutorial 2 Introduction to numerical schemes c 2012 Classifying PDEs Looking at the PDE Au xx + 2Bu xy + Cu yy + Du x + Eu y + Fu +.. = 0, and its discriminant, B 2

More information

MARY ANN HORN be decoupled into three wave equations. Thus, we would hope that results, analogous to those available for the wave equation, would hold

MARY ANN HORN be decoupled into three wave equations. Thus, we would hope that results, analogous to those available for the wave equation, would hold Journal of Mathematical Systems, Estimation, and Control Vol. 8, No. 2, 1998, pp. 1{11 c 1998 Birkhauser-Boston Sharp Trace Regularity for the Solutions of the Equations of Dynamic Elasticity Mary Ann

More information

2 Multidimensional Hyperbolic Problemss where A(u) =f u (u) B(u) =g u (u): (7.1.1c) Multidimensional nite dierence schemes can be simple extensions of

2 Multidimensional Hyperbolic Problemss where A(u) =f u (u) B(u) =g u (u): (7.1.1c) Multidimensional nite dierence schemes can be simple extensions of Chapter 7 Multidimensional Hyperbolic Problems 7.1 Split and Unsplit Dierence Methods Our study of multidimensional parabolic problems in Chapter 5 has laid most of the groundwork for our present task

More information

Abstract. A front tracking method is used to construct weak solutions to

Abstract. A front tracking method is used to construct weak solutions to A Front Tracking Method for Conservation Laws with Boundary Conditions K. Hvistendahl Karlsen, K.{A. Lie, and N. H. Risebro Abstract. A front tracking method is used to construct weak solutions to scalar

More information

Multigrid Approaches to Non-linear Diffusion Problems on Unstructured Meshes

Multigrid Approaches to Non-linear Diffusion Problems on Unstructured Meshes NASA/CR-2001-210660 ICASE Report No. 2001-3 Multigrid Approaches to Non-linear Diffusion Problems on Unstructured Meshes Dimitri J. Mavriplis ICASE, Hampton, Virginia ICASE NASA Langley Research Center

More information

Termination criteria for inexact fixed point methods

Termination criteria for inexact fixed point methods Termination criteria for inexact fixed point methods Philipp Birken 1 October 1, 2013 1 Institute of Mathematics, University of Kassel, Heinrich-Plett-Str. 40, D-34132 Kassel, Germany Department of Mathematics/Computer

More information

1e N

1e N Spectral schemes on triangular elements by Wilhelm Heinrichs and Birgit I. Loch Abstract The Poisson problem with homogeneous Dirichlet boundary conditions is considered on a triangle. The mapping between

More information

1 Introduction We will consider traveling waves for reaction-diusion equations (R-D) u t = nx i;j=1 (a ij (x)u xi ) xj + f(u) uj t=0 = u 0 (x) (1.1) w

1 Introduction We will consider traveling waves for reaction-diusion equations (R-D) u t = nx i;j=1 (a ij (x)u xi ) xj + f(u) uj t=0 = u 0 (x) (1.1) w Reaction-Diusion Fronts in Periodically Layered Media George Papanicolaou and Xue Xin Courant Institute of Mathematical Sciences 251 Mercer Street, New York, N.Y. 10012 Abstract We compute the eective

More information

Waveform Relaxation Method with Toeplitz. Operator Splitting. Sigitas Keras. August Department of Applied Mathematics and Theoretical Physics

Waveform Relaxation Method with Toeplitz. Operator Splitting. Sigitas Keras. August Department of Applied Mathematics and Theoretical Physics UNIVERSITY OF CAMBRIDGE Numerical Analysis Reports Waveform Relaxation Method with Toeplitz Operator Splitting Sigitas Keras DAMTP 1995/NA4 August 1995 Department of Applied Mathematics and Theoretical

More information

NUMERICAL SIMULATION OF INTERACTION BETWEEN INCOMPRESSIBLE FLOW AND AN ELASTIC WALL

NUMERICAL SIMULATION OF INTERACTION BETWEEN INCOMPRESSIBLE FLOW AND AN ELASTIC WALL Proceedings of ALGORITMY 212 pp. 29 218 NUMERICAL SIMULATION OF INTERACTION BETWEEN INCOMPRESSIBLE FLOW AND AN ELASTIC WALL MARTIN HADRAVA, MILOSLAV FEISTAUER, AND PETR SVÁČEK Abstract. The present paper

More information

Blowup for Hyperbolic Equations. Helge Kristian Jenssen and Carlo Sinestrari

Blowup for Hyperbolic Equations. Helge Kristian Jenssen and Carlo Sinestrari Blowup for Hyperbolic Equations Helge Kristian Jenssen and Carlo Sinestrari Abstract. We consider dierent situations of blowup in sup-norm for hyperbolic equations. For scalar conservation laws with a

More information

A Computational Approach to Study a Logistic Equation

A Computational Approach to Study a Logistic Equation Communications in MathematicalAnalysis Volume 1, Number 2, pp. 75 84, 2006 ISSN 0973-3841 2006 Research India Publications A Computational Approach to Study a Logistic Equation G. A. Afrouzi and S. Khademloo

More information

SEMIGROUP APPROACH FOR PARTIAL DIFFERENTIAL EQUATIONS OF EVOLUTION

SEMIGROUP APPROACH FOR PARTIAL DIFFERENTIAL EQUATIONS OF EVOLUTION SEMIGROUP APPROACH FOR PARTIAL DIFFERENTIAL EQUATIONS OF EVOLUTION Istanbul Kemerburgaz University Istanbul Analysis Seminars 24 October 2014 Sabanc University Karaköy Communication Center 1 2 3 4 5 u(x,

More information

quantitative information on the error caused by using the solution of the linear problem to describe the response of the elastic material on a corner

quantitative information on the error caused by using the solution of the linear problem to describe the response of the elastic material on a corner Quantitative Justication of Linearization in Nonlinear Hencky Material Problems 1 Weimin Han and Hong-ci Huang 3 Abstract. The classical linear elasticity theory is based on the assumption that the size

More information

A general theory of discrete ltering. for LES in complex geometry. By Oleg V. Vasilyev AND Thomas S. Lund

A general theory of discrete ltering. for LES in complex geometry. By Oleg V. Vasilyev AND Thomas S. Lund Center for Turbulence Research Annual Research Briefs 997 67 A general theory of discrete ltering for ES in complex geometry By Oleg V. Vasilyev AND Thomas S. und. Motivation and objectives In large eddy

More information

On Nonlinear Dirichlet Neumann Algorithms for Jumping Nonlinearities

On Nonlinear Dirichlet Neumann Algorithms for Jumping Nonlinearities On Nonlinear Dirichlet Neumann Algorithms for Jumping Nonlinearities Heiko Berninger, Ralf Kornhuber, and Oliver Sander FU Berlin, FB Mathematik und Informatik (http://www.math.fu-berlin.de/rd/we-02/numerik/)

More information

On reaching head-to-tail ratios for balanced and unbalanced coins

On reaching head-to-tail ratios for balanced and unbalanced coins Journal of Statistical Planning and Inference 0 (00) 0 0 www.elsevier.com/locate/jspi On reaching head-to-tail ratios for balanced and unbalanced coins Tamas Lengyel Department of Mathematics, Occidental

More information

ERIK STERNER. Discretize the equations in space utilizing a nite volume or nite dierence scheme. 3. Integrate the corresponding time-dependent problem

ERIK STERNER. Discretize the equations in space utilizing a nite volume or nite dierence scheme. 3. Integrate the corresponding time-dependent problem Convergence Acceleration for the Navier{Stokes Equations Using Optimal Semicirculant Approximations Sverker Holmgren y, Henrik Branden y and Erik Sterner y Abstract. The iterative solution of systems of

More information

XIAO-CHUAN CAI AND MAKSYMILIAN DRYJA. strongly elliptic equations discretized by the nite element methods.

XIAO-CHUAN CAI AND MAKSYMILIAN DRYJA. strongly elliptic equations discretized by the nite element methods. Contemporary Mathematics Volume 00, 0000 Domain Decomposition Methods for Monotone Nonlinear Elliptic Problems XIAO-CHUAN CAI AND MAKSYMILIAN DRYJA Abstract. In this paper, we study several overlapping

More information

The Mortar Wavelet Method Silvia Bertoluzza Valerie Perrier y October 29, 1999 Abstract This paper deals with the construction of wavelet approximatio

The Mortar Wavelet Method Silvia Bertoluzza Valerie Perrier y October 29, 1999 Abstract This paper deals with the construction of wavelet approximatio The Mortar Wavelet Method Silvia Bertoluzza Valerie Perrier y October 9, 1999 Abstract This paper deals with the construction of wavelet approximation spaces, in the framework of the Mortar method. We

More information

Splitting of Expanded Tridiagonal Matrices. Seongjai Kim. Abstract. The article addresses a regular splitting of tridiagonal matrices.

Splitting of Expanded Tridiagonal Matrices. Seongjai Kim. Abstract. The article addresses a regular splitting of tridiagonal matrices. Splitting of Expanded Tridiagonal Matrices ga = B? R for Which (B?1 R) = 0 Seongai Kim Abstract The article addresses a regular splitting of tridiagonal matrices. The given tridiagonal matrix A is rst

More information

ON THE ASYMPTOTIC STABILITY IN TERMS OF TWO MEASURES FOR FUNCTIONAL DIFFERENTIAL EQUATIONS. G. Makay

ON THE ASYMPTOTIC STABILITY IN TERMS OF TWO MEASURES FOR FUNCTIONAL DIFFERENTIAL EQUATIONS. G. Makay ON THE ASYMPTOTIC STABILITY IN TERMS OF TWO MEASURES FOR FUNCTIONAL DIFFERENTIAL EQUATIONS G. Makay Student in Mathematics, University of Szeged, Szeged, H-6726, Hungary Key words and phrases: Lyapunov

More information

Bifurcation from the rst eigenvalue of some nonlinear elliptic operators in Banach spaces

Bifurcation from the rst eigenvalue of some nonlinear elliptic operators in Banach spaces Nonlinear Analysis 42 (2000) 561 572 www.elsevier.nl/locate/na Bifurcation from the rst eigenvalue of some nonlinear elliptic operators in Banach spaces Pavel Drabek a;, Nikos M. Stavrakakis b a Department

More information

Stability of implicit extrapolation methods. Abstract. Multilevel methods are generally based on a splitting of the solution

Stability of implicit extrapolation methods. Abstract. Multilevel methods are generally based on a splitting of the solution Contemporary Mathematics Volume 00, 0000 Stability of implicit extrapolation methods Abstract. Multilevel methods are generally based on a splitting of the solution space associated with a nested sequence

More information

Introduction to numerical schemes

Introduction to numerical schemes 236861 Numerical Geometry of Images Tutorial 2 Introduction to numerical schemes Heat equation The simple parabolic PDE with the initial values u t = K 2 u 2 x u(0, x) = u 0 (x) and some boundary conditions

More information

Unconstrained optimization

Unconstrained optimization Chapter 4 Unconstrained optimization An unconstrained optimization problem takes the form min x Rnf(x) (4.1) for a target functional (also called objective function) f : R n R. In this chapter and throughout

More information

Bifurcation analysis of incompressible ow in a driven cavity F.W. Wubs y, G. Tiesinga z and A.E.P. Veldman x Abstract Knowledge of the transition point of steady to periodic ow and the frequency occurring

More information

k=6, t=100π, solid line: exact solution; dashed line / squares: numerical solution

k=6, t=100π, solid line: exact solution; dashed line / squares: numerical solution DIFFERENT FORMULATIONS OF THE DISCONTINUOUS GALERKIN METHOD FOR THE VISCOUS TERMS CHI-WANG SHU y Abstract. Discontinuous Galerkin method is a nite element method using completely discontinuous piecewise

More information

Partial Differential Equations

Partial Differential Equations Partial Differential Equations Introduction Deng Li Discretization Methods Chunfang Chen, Danny Thorne, Adam Zornes CS521 Feb.,7, 2006 What do You Stand For? A PDE is a Partial Differential Equation This

More information

Math 411 Preliminaries

Math 411 Preliminaries Math 411 Preliminaries Provide a list of preliminary vocabulary and concepts Preliminary Basic Netwon s method, Taylor series expansion (for single and multiple variables), Eigenvalue, Eigenvector, Vector

More information

AM 205: lecture 19. Last time: Conditions for optimality, Newton s method for optimization Today: survey of optimization methods

AM 205: lecture 19. Last time: Conditions for optimality, Newton s method for optimization Today: survey of optimization methods AM 205: lecture 19 Last time: Conditions for optimality, Newton s method for optimization Today: survey of optimization methods Quasi-Newton Methods General form of quasi-newton methods: x k+1 = x k α

More information

Maria Cameron. f(x) = 1 n

Maria Cameron. f(x) = 1 n Maria Cameron 1. Local algorithms for solving nonlinear equations Here we discuss local methods for nonlinear equations r(x) =. These methods are Newton, inexact Newton and quasi-newton. We will show that

More information

Pointwise convergence rate for nonlinear conservation. Eitan Tadmor and Tao Tang

Pointwise convergence rate for nonlinear conservation. Eitan Tadmor and Tao Tang Pointwise convergence rate for nonlinear conservation laws Eitan Tadmor and Tao Tang Abstract. We introduce a new method to obtain pointwise error estimates for vanishing viscosity and nite dierence approximations

More information

Classification of partial differential equations and their solution characteristics

Classification of partial differential equations and their solution characteristics 9 TH INDO GERMAN WINTER ACADEMY 2010 Classification of partial differential equations and their solution characteristics By Ankita Bhutani IIT Roorkee Tutors: Prof. V. Buwa Prof. S. V. R. Rao Prof. U.

More information

The iterative convex minorant algorithm for nonparametric estimation

The iterative convex minorant algorithm for nonparametric estimation The iterative convex minorant algorithm for nonparametric estimation Report 95-05 Geurt Jongbloed Technische Universiteit Delft Delft University of Technology Faculteit der Technische Wiskunde en Informatica

More information

Scientific Computing: An Introductory Survey

Scientific Computing: An Introductory Survey Scientific Computing: An Introductory Survey Chapter 5 Nonlinear Equations Prof. Michael T. Heath Department of Computer Science University of Illinois at Urbana-Champaign Copyright c 2002. Reproduction

More information

Lecture Introduction

Lecture Introduction Lecture 1 1.1 Introduction The theory of Partial Differential Equations (PDEs) is central to mathematics, both pure and applied. The main difference between the theory of PDEs and the theory of Ordinary

More information

A memory-ecient nite element method for systems of reaction diusion equations with non-smooth forcing

A memory-ecient nite element method for systems of reaction diusion equations with non-smooth forcing Available online at www.sciencedirect.com Journal of Computational and Applied Mathematics 169 (2004) 431 458 www.elsevier.com/locate/cam A memory-ecient nite element method for systems of reaction diusion

More information

Contents. 2.1 Vectors in R n. Linear Algebra (part 2) : Vector Spaces (by Evan Dummit, 2017, v. 2.50) 2 Vector Spaces

Contents. 2.1 Vectors in R n. Linear Algebra (part 2) : Vector Spaces (by Evan Dummit, 2017, v. 2.50) 2 Vector Spaces Linear Algebra (part 2) : Vector Spaces (by Evan Dummit, 2017, v 250) Contents 2 Vector Spaces 1 21 Vectors in R n 1 22 The Formal Denition of a Vector Space 4 23 Subspaces 6 24 Linear Combinations and

More information

2 ZHANGXIN CHEN AND RICHARD EWING dierentiating the resulting pressure and multiplying it by the rough coecient. This approach generates a rough and i

2 ZHANGXIN CHEN AND RICHARD EWING dierentiating the resulting pressure and multiplying it by the rough coecient. This approach generates a rough and i FROM SINGLE-PHASE TO COMPOSITIONAL FLOW: APPLICABILITY OF MIXED FINITE ELEMENTS Zhangxin Chen and Richard E. Ewing Abstract. In this paper we discuss the formulation of the governing equations that describe

More information

PARAMETER IDENTIFICATION IN THE FREQUENCY DOMAIN. H.T. Banks and Yun Wang. Center for Research in Scientic Computation

PARAMETER IDENTIFICATION IN THE FREQUENCY DOMAIN. H.T. Banks and Yun Wang. Center for Research in Scientic Computation PARAMETER IDENTIFICATION IN THE FREQUENCY DOMAIN H.T. Banks and Yun Wang Center for Research in Scientic Computation North Carolina State University Raleigh, NC 7695-805 Revised: March 1993 Abstract In

More information

Chapter Two: Numerical Methods for Elliptic PDEs. 1 Finite Difference Methods for Elliptic PDEs

Chapter Two: Numerical Methods for Elliptic PDEs. 1 Finite Difference Methods for Elliptic PDEs Chapter Two: Numerical Methods for Elliptic PDEs Finite Difference Methods for Elliptic PDEs.. Finite difference scheme. We consider a simple example u := subject to Dirichlet boundary conditions ( ) u

More information

Math 1270 Honors ODE I Fall, 2008 Class notes # 14. x 0 = F (x; y) y 0 = G (x; y) u 0 = au + bv = cu + dv

Math 1270 Honors ODE I Fall, 2008 Class notes # 14. x 0 = F (x; y) y 0 = G (x; y) u 0 = au + bv = cu + dv Math 1270 Honors ODE I Fall, 2008 Class notes # 1 We have learned how to study nonlinear systems x 0 = F (x; y) y 0 = G (x; y) (1) by linearizing around equilibrium points. If (x 0 ; y 0 ) is an equilibrium

More information

Operator Dependent Interpolation. A. Krechel. K. Stuben. German National Research Center for Information Technology (GMD)

Operator Dependent Interpolation. A. Krechel. K. Stuben. German National Research Center for Information Technology (GMD) Operator Dependent Interpolation in Algebraic Multigrid A. Krechel K. Stuben German National Research Center for Information Technology (GMD) Institute for Algorithms and Scientic Computing (SCAI) Schloss

More information

1. Nonlinear Equations. This lecture note excerpted parts from Michael Heath and Max Gunzburger. f(x) = 0

1. Nonlinear Equations. This lecture note excerpted parts from Michael Heath and Max Gunzburger. f(x) = 0 Numerical Analysis 1 1. Nonlinear Equations This lecture note excerpted parts from Michael Heath and Max Gunzburger. Given function f, we seek value x for which where f : D R n R n is nonlinear. f(x) =

More information

Memoirs on Dierential Equations and Mathematical Physics

Memoirs on Dierential Equations and Mathematical Physics Memoirs on Dierential Equations and Mathematical Physics Volume 20, 2000, 113{126 T. Chantladze, N. Kandelaki, A. Lomtatidze, and D. Ugulava THE P-LAPLACIAN AND CONNECTED PAIRS OF FUNCTIONS Abstract. The

More information

Journal of Computational and Applied Mathematics. Determination of a material constant in the impedance boundary condition for electromagnetic fields

Journal of Computational and Applied Mathematics. Determination of a material constant in the impedance boundary condition for electromagnetic fields Journal of Computational and Applied Mathematics 234 (2010) 2062 2068 Contents lists available at ScienceDirect Journal of Computational and Applied Mathematics journal homepage: www.elsevier.com/locate/cam

More information

The WENO Method for Non-Equidistant Meshes

The WENO Method for Non-Equidistant Meshes The WENO Method for Non-Equidistant Meshes Philip Rupp September 11, 01, Berlin Contents 1 Introduction 1.1 Settings and Conditions...................... The WENO Schemes 4.1 The Interpolation Problem.....................

More information

SOE3213/4: CFD Lecture 3

SOE3213/4: CFD Lecture 3 CFD { SOE323/4: CFD Lecture 3 @u x @t @u y @t @u z @t r:u = 0 () + r:(uu x ) = + r:(uu y ) = + r:(uu z ) = @x @y @z + r 2 u x (2) + r 2 u y (3) + r 2 u z (4) Transport equation form, with source @x Two

More information

OPERATOR SPLITTING METHODS FOR SYSTEMS OF CONVECTION-DIFFUSION EQUATIONS: NONLINEAR ERROR MECHANISMS AND CORRECTION STRATEGIES

OPERATOR SPLITTING METHODS FOR SYSTEMS OF CONVECTION-DIFFUSION EQUATIONS: NONLINEAR ERROR MECHANISMS AND CORRECTION STRATEGIES OPERATOR SPLITTING METHODS FOR SYSTEMS OF CONVECTION-DIFFUSION EQUATIONS: NONLINEAR ERROR MECHANISMS AND CORRECTION STRATEGIES K. HVISTENDAHL KARLSEN a, K.{A. LIE b;c, J. R. NATVIG c, H. F. NORDHAUG a,

More information

1 Newton s Method. Suppose we want to solve: x R. At x = x, f (x) can be approximated by:

1 Newton s Method. Suppose we want to solve: x R. At x = x, f (x) can be approximated by: Newton s Method Suppose we want to solve: (P:) min f (x) At x = x, f (x) can be approximated by: n x R. f (x) h(x) := f ( x)+ f ( x) T (x x)+ (x x) t H ( x)(x x), 2 which is the quadratic Taylor expansion

More information

Two-parameter regularization method for determining the heat source

Two-parameter regularization method for determining the heat source Global Journal of Pure and Applied Mathematics. ISSN 0973-1768 Volume 13, Number 8 (017), pp. 3937-3950 Research India Publications http://www.ripublication.com Two-parameter regularization method for

More information

Average Reward Parameters

Average Reward Parameters Simulation-Based Optimization of Markov Reward Processes: Implementation Issues Peter Marbach 2 John N. Tsitsiklis 3 Abstract We consider discrete time, nite state space Markov reward processes which depend

More information

Identification of a memory kernel in a nonlinear parabolic integro-differential problem

Identification of a memory kernel in a nonlinear parabolic integro-differential problem FACULTY OF ENGINEERING AND ARCHITECTURE Identification of a memory kernel in a nonlinear parabolic integro-differential problem K. Van Bockstal, M. Slodička and F. Gistelinck Ghent University Department

More information

HETEROGENEOUS MULTISCALE METHOD IN EDDY CURRENTS MODELING

HETEROGENEOUS MULTISCALE METHOD IN EDDY CURRENTS MODELING Proceedings of ALGORITMY 2009 pp. 219 225 HETEROGENEOUS MULTISCALE METHOD IN EDDY CURRENTS MODELING JÁN BUŠA, JR. AND VALDEMAR MELICHER Abstract. The induction of eddy currents in a conductive piece is

More information

LECTURE # 0 BASIC NOTATIONS AND CONCEPTS IN THE THEORY OF PARTIAL DIFFERENTIAL EQUATIONS (PDES)

LECTURE # 0 BASIC NOTATIONS AND CONCEPTS IN THE THEORY OF PARTIAL DIFFERENTIAL EQUATIONS (PDES) LECTURE # 0 BASIC NOTATIONS AND CONCEPTS IN THE THEORY OF PARTIAL DIFFERENTIAL EQUATIONS (PDES) RAYTCHO LAZAROV 1 Notations and Basic Functional Spaces Scalar function in R d, d 1 will be denoted by u,

More information

Time Integration Methods for the Heat Equation

Time Integration Methods for the Heat Equation Time Integration Methods for the Heat Equation Tobias Köppl - JASS March 2008 Heat Equation: t u u = 0 Preface This paper is a short summary of my talk about the topic: Time Integration Methods for the

More information

13 PDEs on spatially bounded domains: initial boundary value problems (IBVPs)

13 PDEs on spatially bounded domains: initial boundary value problems (IBVPs) 13 PDEs on spatially bounded domains: initial boundary value problems (IBVPs) A prototypical problem we will discuss in detail is the 1D diffusion equation u t = Du xx < x < l, t > finite-length rod u(x,

More information

Outline Introduction: Problem Description Diculties Algebraic Structure: Algebraic Varieties Rank Decient Toeplitz Matrices Constructing Lower Rank St

Outline Introduction: Problem Description Diculties Algebraic Structure: Algebraic Varieties Rank Decient Toeplitz Matrices Constructing Lower Rank St Structured Lower Rank Approximation by Moody T. Chu (NCSU) joint with Robert E. Funderlic (NCSU) and Robert J. Plemmons (Wake Forest) March 5, 1998 Outline Introduction: Problem Description Diculties Algebraic

More information

Optimization Methods. Lecture 18: Optimality Conditions and. Gradient Methods. for Unconstrained Optimization

Optimization Methods. Lecture 18: Optimality Conditions and. Gradient Methods. for Unconstrained Optimization 5.93 Optimization Methods Lecture 8: Optimality Conditions and Gradient Methods for Unconstrained Optimization Outline. Necessary and sucient optimality conditions Slide. Gradient m e t h o d s 3. The

More information

Scientific Computing: An Introductory Survey

Scientific Computing: An Introductory Survey Scientific Computing: An Introductory Survey Chapter 11 Partial Differential Equations Prof. Michael T. Heath Department of Computer Science University of Illinois at Urbana-Champaign Copyright c 2002.

More information

Weak solutions for some quasilinear elliptic equations by the sub-supersolution method

Weak solutions for some quasilinear elliptic equations by the sub-supersolution method Nonlinear Analysis 4 (000) 995 100 www.elsevier.nl/locate/na Weak solutions for some quasilinear elliptic equations by the sub-supersolution method Manuel Delgado, Antonio Suarez Departamento de Ecuaciones

More information

Iterative procedure for multidimesional Euler equations Abstracts A numerical iterative scheme is suggested to solve the Euler equations in two and th

Iterative procedure for multidimesional Euler equations Abstracts A numerical iterative scheme is suggested to solve the Euler equations in two and th Iterative procedure for multidimensional Euler equations W. Dreyer, M. Kunik, K. Sabelfeld, N. Simonov, and K. Wilmanski Weierstra Institute for Applied Analysis and Stochastics Mohrenstra e 39, 07 Berlin,

More information

Lecture 1: Dynamic Programming

Lecture 1: Dynamic Programming Lecture 1: Dynamic Programming Fatih Guvenen November 2, 2016 Fatih Guvenen Lecture 1: Dynamic Programming November 2, 2016 1 / 32 Goal Solve V (k, z) =max c,k 0 u(c)+ E(V (k 0, z 0 ) z) c + k 0 =(1 +

More information

MULTIGRID PRECONDITIONING FOR THE BIHARMONIC DIRICHLET PROBLEM M. R. HANISCH

MULTIGRID PRECONDITIONING FOR THE BIHARMONIC DIRICHLET PROBLEM M. R. HANISCH MULTIGRID PRECONDITIONING FOR THE BIHARMONIC DIRICHLET PROBLEM M. R. HANISCH Abstract. A multigrid preconditioning scheme for solving the Ciarlet-Raviart mixed method equations for the biharmonic Dirichlet

More information

5 Handling Constraints

5 Handling Constraints 5 Handling Constraints Engineering design optimization problems are very rarely unconstrained. Moreover, the constraints that appear in these problems are typically nonlinear. This motivates our interest

More information

A Finite Element Method for an Ill-Posed Problem. Martin-Luther-Universitat, Fachbereich Mathematik/Informatik,Postfach 8, D Halle, Abstract

A Finite Element Method for an Ill-Posed Problem. Martin-Luther-Universitat, Fachbereich Mathematik/Informatik,Postfach 8, D Halle, Abstract A Finite Element Method for an Ill-Posed Problem W. Lucht Martin-Luther-Universitat, Fachbereich Mathematik/Informatik,Postfach 8, D-699 Halle, Germany Abstract For an ill-posed problem which has its origin

More information

PDE Solvers for Fluid Flow

PDE Solvers for Fluid Flow PDE Solvers for Fluid Flow issues and algorithms for the Streaming Supercomputer Eran Guendelman February 5, 2002 Topics Equations for incompressible fluid flow 3 model PDEs: Hyperbolic, Elliptic, Parabolic

More information

Numerical solution of fourth order parabolic partial dierential equation using parametric septic splines

Numerical solution of fourth order parabolic partial dierential equation using parametric septic splines Hacettepe Journal of Mathematics and Statistics Volume 5 20, 07 082 Numerical solution of fourth order parabolic partial dierential equation using parametric septic splines Arshad Khan and Talat Sultana

More information

and P RP k = gt k (g k? g k? ) kg k? k ; (.5) where kk is the Euclidean norm. This paper deals with another conjugate gradient method, the method of s

and P RP k = gt k (g k? g k? ) kg k? k ; (.5) where kk is the Euclidean norm. This paper deals with another conjugate gradient method, the method of s Global Convergence of the Method of Shortest Residuals Yu-hong Dai and Ya-xiang Yuan State Key Laboratory of Scientic and Engineering Computing, Institute of Computational Mathematics and Scientic/Engineering

More information

ADDITIVE SCHWARZ FOR SCHUR COMPLEMENT 305 the parallel implementation of both preconditioners on distributed memory platforms, and compare their perfo

ADDITIVE SCHWARZ FOR SCHUR COMPLEMENT 305 the parallel implementation of both preconditioners on distributed memory platforms, and compare their perfo 35 Additive Schwarz for the Schur Complement Method Luiz M. Carvalho and Luc Giraud 1 Introduction Domain decomposition methods for solving elliptic boundary problems have been receiving increasing attention

More information

A FAST SOLVER FOR ELLIPTIC EQUATIONS WITH HARMONIC COEFFICIENT APPROXIMATIONS

A FAST SOLVER FOR ELLIPTIC EQUATIONS WITH HARMONIC COEFFICIENT APPROXIMATIONS Proceedings of ALGORITMY 2005 pp. 222 229 A FAST SOLVER FOR ELLIPTIC EQUATIONS WITH HARMONIC COEFFICIENT APPROXIMATIONS ELENA BRAVERMAN, MOSHE ISRAELI, AND ALEXANDER SHERMAN Abstract. Based on a fast subtractional

More information

Numerical solutions of nonlinear systems of equations

Numerical solutions of nonlinear systems of equations Numerical solutions of nonlinear systems of equations Tsung-Ming Huang Department of Mathematics National Taiwan Normal University, Taiwan E-mail: min@math.ntnu.edu.tw August 28, 2011 Outline 1 Fixed points

More information

The solution of the discretized incompressible Navier-Stokes equations with iterative methods

The solution of the discretized incompressible Navier-Stokes equations with iterative methods The solution of the discretized incompressible Navier-Stokes equations with iterative methods Report 93-54 C. Vuik Technische Universiteit Delft Delft University of Technology Faculteit der Technische

More information

system of PenroseFife type Olaf Klein Weierstrass Institute for Applied Analysis and Stochastics Mohrenstrasse 39 D10117 Berlin Germany

system of PenroseFife type Olaf Klein Weierstrass Institute for Applied Analysis and Stochastics Mohrenstrasse 39 D10117 Berlin Germany A class of time discrete schemes for a phaseeld system of PenroseFife type Olaf Klein Weierstrass Institute for Applied Analysis and Stochastics Mohrenstrasse 39 D7 Berlin Germany email: klein@wias-berlin.de

More information

Elliptic Problems / Multigrid. PHY 604: Computational Methods for Physics and Astrophysics II

Elliptic Problems / Multigrid. PHY 604: Computational Methods for Physics and Astrophysics II Elliptic Problems / Multigrid Summary of Hyperbolic PDEs We looked at a simple linear and a nonlinear scalar hyperbolic PDE There is a speed associated with the change of the solution Explicit methods

More information

AM 205: lecture 19. Last time: Conditions for optimality Today: Newton s method for optimization, survey of optimization methods

AM 205: lecture 19. Last time: Conditions for optimality Today: Newton s method for optimization, survey of optimization methods AM 205: lecture 19 Last time: Conditions for optimality Today: Newton s method for optimization, survey of optimization methods Optimality Conditions: Equality Constrained Case As another example of equality

More information

ETNA Kent State University

ETNA Kent State University Electronic Transactions on Numerical Analysis. Volume 4, pp. 373-38, 23. Copyright 23,. ISSN 68-963. ETNA A NOTE ON THE RELATION BETWEEN THE NEWTON HOMOTOPY METHOD AND THE DAMPED NEWTON METHOD XUPING ZHANG

More information

Numerical Methods for Large-Scale Nonlinear Systems

Numerical Methods for Large-Scale Nonlinear Systems Numerical Methods for Large-Scale Nonlinear Systems Handouts by Ronald H.W. Hoppe following the monograph P. Deuflhard Newton Methods for Nonlinear Problems Springer, Berlin-Heidelberg-New York, 2004 Num.

More information

New Lyapunov Krasovskii functionals for stability of linear retarded and neutral type systems

New Lyapunov Krasovskii functionals for stability of linear retarded and neutral type systems Systems & Control Letters 43 (21 39 319 www.elsevier.com/locate/sysconle New Lyapunov Krasovskii functionals for stability of linear retarded and neutral type systems E. Fridman Department of Electrical

More information

PHYS 410/555 Computational Physics Solution of Non Linear Equations (a.k.a. Root Finding) (Reference Numerical Recipes, 9.0, 9.1, 9.

PHYS 410/555 Computational Physics Solution of Non Linear Equations (a.k.a. Root Finding) (Reference Numerical Recipes, 9.0, 9.1, 9. PHYS 410/555 Computational Physics Solution of Non Linear Equations (a.k.a. Root Finding) (Reference Numerical Recipes, 9.0, 9.1, 9.4) We will consider two cases 1. f(x) = 0 1-dimensional 2. f(x) = 0 d-dimensional

More information

Line Search Methods for Unconstrained Optimisation

Line Search Methods for Unconstrained Optimisation Line Search Methods for Unconstrained Optimisation Lecture 8, Numerical Linear Algebra and Optimisation Oxford University Computing Laboratory, MT 2007 Dr Raphael Hauser (hauser@comlab.ox.ac.uk) The Generic

More information

2 IVAN YOTOV decomposition solvers and preconditioners are described in Section 3. Computational results are given in Section Multibloc formulat

2 IVAN YOTOV decomposition solvers and preconditioners are described in Section 3. Computational results are given in Section Multibloc formulat INTERFACE SOLVERS AND PRECONDITIONERS OF DOMAIN DECOMPOSITION TYPE FOR MULTIPHASE FLOW IN MULTIBLOCK POROUS MEDIA IVAN YOTOV Abstract. A multibloc mortar approach to modeling multiphase ow in porous media

More information