LECTURE # 0 BASIC NOTATIONS AND CONCEPTS IN THE THEORY OF PARTIAL DIFFERENTIAL EQUATIONS (PDES)
|
|
- Gabriel Edwards
- 5 years ago
- Views:
Transcription
1 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, v, z, f etc Vector functions in R d will be denoted by u, b etc with components u i and b i, i = 1,, d, correspondingly Also we shall use the standard notation for a dot (scalar) product of two vector functions u and b: u b = u 1 b 1 + u 2 b u d b d and the operator = ( x 1,, x d ) We shall use the following widely accepted notations for the differential operators used in this class (see, eg [1, 3]): (1) Gradient of a scalar function u: ( ) u :=,, u = (u x1,, u xd ), (obviously this is a vector); x 1 x d Here an further u xi denotes u x i (2) Divergence of a vector function u = (u 1,, u d ) u := u 1,x1 + + u d,xd ; divergence of a vector function is a scalar function (3) Laplace s operator u = ( u) = u x1 x u xd x d Further, in we shall consider various function spaces: (1) C() = {u : R u is continuous on }; (2) C k () = {u : R u is k-times continuously differentiable in }; (3) C () = {u : R u is infinitely differentiable in }; (4) C 0 (), C0 k(), C 0 () the above classes of function with compact support in (5) For 1 p < L p () = {u : R u is Lebesgue measurable u L p () < }, where ( ) 1 u L p () = u p p dx We say that u L 1 loc () if u L 1 (D) = D u dx < for any subdomain D Further, we introduce the notion of generalized (weak) derivative of u We say that xi u = u x i L 1 loc () is a weak derivative of u if xi uφdx = u xi φ dx, φ C0 () Date: April 2,
2 2 RAYTCHO LAZAROV Example: Show that for = (0, 1) { 0 x < 1 u(x) = 2 x 1 2 x > 1 2 has weak derivative u (x) = { 0 x < x > 1 2 Now we define the Sobolev space W 1,p () as set of functions u L p () having their weak derivatives of first order in L p () We shall work with the Hilbert space H 1 () := W 1,2 () Then the norm in H 1 () is defined by Further, we shall use the space u 2 H 1 = u 2 L 2 () + u 2 L 2 () H 1 0 () = {u H 1 () : u(x) = 0, x } Obviously, H 1 () is a subspace of L 2 () We shall further use two important properties for these two spaces (a) Poincaré-Friedrichs inequality: there is a constnat c 0 > 0, dependent only on the domain, such that u H 1 0 () : c 0 u 2 L 2 () u 2 L 2 (), u H 1 () : c 0 u 2 L 2 () u 2 L 2 () + ( udx) 2 (b) H 1 () is compactly embedded in L 2 (): this means that from any bounded in L 2 () sequence one can extract a convergent in H0 1 () subsequence; In a similar way one can define higher order weak derivatives and Sobolev spaces of any integer order k To introduce these we need some notations for higher order partial derivatives Denote by α = (α 1,, α d ) a multi-index and α i 0 integer numbers, α = α α d Then D α α u 2 u u := x α 1 1, eg = D α u, α = (1, 1) xα d d x 1 x 2 Then the weak α derivative of u is defined as D α uφdx = ( 1) α u D α φ dx, φ C0 () Now we define the Sobolev space W k,p () as set of functions u L p () having their weak derivatives of order k in L p () We shall work with the Hilbert space H k () := W k,2 () Then the norm in H k () is defined by u 2 H = D α u 2 k L 2 () α k We shall use the following basic result, called Sobolev lemma: Lemma 1 (Sobolev s lemma, [5, more general form on page 107]) Assume that the region has Lipshitz boundary and k > d/2 Then the elements of H k () are bounded continuous functions on and H k () is continuously embedded in C( )
3 LECTURE # 0: BASIC NOTATIONS AND CONCEPTS 3 In the modern numerical analysis related to PDEs important role play the notion of weak solutions of the differential equation Now we formulated various PDEs in their strong form Usually, this assumes that all derivatives that are involved in the PDE are continuous (or piecewise continuous) functions in the domain 2 Basic Differential Equations (in strong form) We shall consider various problems related to the following PDEs in a bounded domain R d A list of all these equations and many more could be found in the book of L Evans [1, p 3-6], which I recommend as a basic advanced textbook on PDEs (1) Poisson Equation for u = u(x): u = f, x Appropriate boundary conditions for this equations are: Dirichlet: u = g on the boundary ; Neumann: u n = g on the boundary ; Robin type: u n + u = g on the boundary Here n is the outer unit normal vector to and g is a given function on (2) Helmholtz Equation for u = u(x): u ω 2 u = f, x, ω a real number, with appropriate boundary conditions (see, eg Poisson equation) (3) Second Order Elliptic Equation (diffusion-reaction): (K(x) u) + c(x)u = f, x with appropriate boundary conditions Here K(x) is a symmetric matrix that is uniformly positive definite in, ie there is a constant α 0 > 0 such that ξ T K(x)ξ α 0 ξ 2 for all ξ R d and x and c(x) 0 in The functions K(x), c(x) and f(x) are given on Remark: The one-dimensional variant (ie all functions depend only on the single variable x) of this equation is (k(x)u ) + c(x)u = f(x), x (0, 1) with relevant boundary conditions (4) Mixed Form of the Diffusion-Reaction Equation: Introduce the flux dependent variable q = K(x) u so that the diffusion-reaction equation reads as q + c(x)u = f This first order system is called mixed form of the equation (5) Diffusion-Convection-Reaction Equation: (K(x) u + bu) + c(x)u = f, x Proper boundary conditions for this equation are in general more complicated One simple BC is Dirichlet BC (6) Transient (Time-Dependent) Heat Equation: Here the unknown function depends on x and time t > 0, ie u(x, t): u t u = f, x, t > 0 Together with proper BC we give also an initial condition (IC): u(x, 0) = u 0 (x), where u 0 (x) is a given function in
4 4 RAYTCHO LAZAROV (7) Second Order Hyperbolic Equation for u = u(x, t): u tt u = f, x, t > 0, This equation describes the vibrations of a thin membrane under the transverse load f(x, t) and the initial displacement u(x, 0) = u 0 (x) and initial velocity u t (x, 0) = u 1 (x) for x Appropriate BC have to be specified as well (8) Transport Equation for u = u(x, t): u t + b u + c(x)u = f, x, t > 0 with standard initial condition u(x, 0) = u 0 (x) and BC given on the inflow part of the boundary, Γ in, namely, u n = g, x Γ in, t > 0, Γ in := {x : b(x) n(x) < 0} Sometimes it is better to have this in a conservative form u t + (b u) + c(x)u = f, x, t > 0 with the steady state case (b u) + c(x)u = f (9) Nonlinear Conservation Law for u = u(x, t): u t + b(u) = f, x R, t > 0, where b(u) is a convex function of u, eg Burger s equation for d = 1, b(u) = u 2 (10) Nonlinear Poisson Equation for u = u(x): (11) p-laplacian Equation for u = u(x): u = f(u), x ( u p 1 u) = 0, x with appropriate boundary conditions (12) Minimal Surface Equation for u = u(x): ( ) u = 0, x (1 + u 2 ) 1 2 (13) Stokes System: This is a system for the fluid velocity u and pressure p in a bounded domain : u + p = f, u = 0, with Dirichlet boundary condition for u and no BC for p (14) Equations of linear elasticity The primary dependent variable is the displacement vector u(x) = (u 1,, u d ) that describes the deformations of an elastic body occupying a volume The gradient of u is the tensor u 1 x 1 u := u d x 1 u 1 x d u d x d, ɛ = 1 2 ( u + ( u)t ), σ = Kɛ
5 LECTURE # 0: BASIC NOTATIONS AND CONCEPTS 5 Here ɛ is the strain tensor, σ is the stress tensor and K is a matrix giving the relation between the strains and the stresses (elements of the matrix depend on the Young and Poisson moduli) The equation then is σ = f x, is taken row-wise from σ 3 Boundary Value Problems Is is well known that in order to select a unique solution of a given PDE we need to specify some boundary conditions (BC) or/and initial conditions (IC) Eg the equation u = 1 has infinitely many solutions u(x) = 1 2 x2 + c 1 x + c 2 depending on two arbitrary constants c 1 and c 2 However, the boundary value problem u = 1, x (0, 1), u(0) = u(1) = 0 has unique solution u(x) = 1 2x(x 1) One of the most common boundary conditions are Dirichlet BC u(x) = g(x), and/or Neumann BC, u n = g(x), x Γ, where g(x) is a given function and n is the outer normal unit vector to the boundary Γ Another possibility is to have Dirichlet BC on Γ D and Neumann BC on Γ N where Γ = Γ D Γ N We shall consider differential equations and boundary conditions that constitute a well posed problem Loosely speaking, this is a problem that has unique solution and the solution depends continuously on the data in certain sense (1) 4 The notion of weak form of a boundary value problem (BVP) (a) We shall consider the following boundary value problem: find u(x) such that (K(x) u) + c(x)u = f for x and u(x) = 0 for x Γ We multiply this equation by a test function v(x) (a smooth function with compact support in the domain ) and integrate over the domain to get: ( (K(x) u) + c(x)u)v dx = fv dx After applying the Stokes theorem in the first term on the left and assuming v = 0 on Γ we get ( (K(x) u) v dx = K(x) u v dx K(x) u vds Γ = K(x) u v dx Thus, the solution u(x) satisfies the following integral identity: ( ) K(x) u v + c(x)uv dx = fv dx v H0 1 () Now if K(x) is smooth (in fact it is enough to be measurable and bounded in ) then the above integral identity is well defined for u, v H 1 0 () To simplify the notation and to reduce some extensive writing the weak formulation of the BVP can be written as (2) Find u H 1 0 () such that a(u, v) = l(v) v H 1 0 (), where the bilinear form a(u, v) and the linear from l(v) are defined by ( ) (3) a(u, v) := K(x) u v + c(x)u v dx and l(v) := fv dx
6 6 RAYTCHO LAZAROV Note, that u H0 1 () implies that the solution u(x) = 0 on the boundary Γ This boundary condition is often called essential boundary condition (b) Consider the same problem with nonhomegeneous Dirichlet data in a mixed form: q + K(x) u = 0, q + c(x)u = f, x, u(x) = g(x), x Γ Rewrite the first equation in the form K 1 (x)q + u = 0, multiply the last one by a test vector-function r, integrate over, apply the Stokes theorem to the second term and take into account the boundary condition to get K 1 (x)q rdx u r dx + g r nds = 0 Further, we multiply the second equation by a test function v and integrate over to get q vdx + c(x)uv dx = fv dx Obviously, these integral identities make sense for vector fields such that the divergence q exists in a weak sense and belongs to L 2 () Now we define the space This space is endowed with the norm H(div; ) = {r L 2 () d : r L 2 ()} r 2 H div = r 2 L 2 + r 2 L 2 Then the weak form of the mixed system is: find q H(div) and u L 2 such that the integral identity (K 1 (x)q, r) (u, r) ( q, v) (cu, v) = g, r n (f, v) is satisfied for all r H(div) and v L 2 Here we have used the shorthand notation K 1 (x)q rdx = (K 1 (x)q, r), fv dx = (f, v) and gvds = g, v Note that the solutions q and u are not required to satisfy any boundary conditions The boundary conditions have become part of the weak formulation, ie they are included in the weak form Such boundary conditions are called natural BC for the setting 5 Linear Partial Differential Equations of Second Order All equations considered above are linear This means that if we introduce the operator, for example Lu := (K(x) u + bu) + c(x)u then it satisfies the relation L(u + v) = Lu + Lv In more general cases we consider the linear differential operator d 2 u d u Lu := K ij (x) + b i, x R d x i x j x i i,j=1 with given coefficients K ij and b In analogy with the classification of conic sections, at point x the differential equation is called elliptic, if the eigenvalues of the matrix K(x) with entries K ij are not zero and have the same sign; i=1 Γ Γ
7 LECTURE # 0: BASIC NOTATIONS AND CONCEPTS 7 hyperbolic, if one eigenvalue is positive and the others are all negative (or if one eigenvalue is negative and the others are all positive); parabolic, if exactly one eigenvalue is equal to 0 6 Exercises: Problem 1: (see [4]) Decide whether the operator is linear or nonlinear Lu := u x1 x 1 + x 1 u x2, u = u(x), x R 2, Lu := u tt u xx + u 2, u = u(x, t), x R, Lu := u t + u xx u, u = u(x, t), x R Problem 2: (see [4]) Determine the type of the differential operator (x R 2 ): Lu := u x1 x 1 u x1 x 2 + 2u x2 + u x2 x 2 3u x2 x 1 + 4u; Lu := 3u x2 + u x1 x 2 Lu := 9u x1 x 1 + 6u x1 x 2 + u x2 x 2 Problem 3: Consider the following two-points boundary value second order problem in 1-D: Find a function u defined ae in ]0, 1[ such that (4) ( xk(x)u (x) ) + xc(x)u(x) = xf(x) ae in ]0, 1[, ( xu (x) ) = 0 and K(1)u (1) + u(1) = 0, lim x 0 where K C 1 ([0, 1]), q C 0 ([0, 1]) and f L 2 (0, 1) are given functions Assume that there exists a constant κ 0 > 0 such that K(x) κ 0 and c(x) 0 for all x [0, 1] (1) Assume that the Sobolev space V is obtained by completion of the continuously differentiable functions in ]0, 1[, C 1 ([0, 1]), with respect to the norm ( v V = 1/2 xv 2 L 2 (0,1) + xv 2 L (0,1)) 2 Derive the weak form of the problem (4) in the space V (2) Prove that the corresponding bilinear from of the weak form of (4) is coercive in V Problem 4: Consider the boundary value problem u (4) = f(x), 0 < x < 1, u(0) = 0, u (0) = 0, u (1) + u (1) = β, u (1) = γ, where f(x) is a given function on (0, 1) and β and γ are given constants (1) Give the weak formulation of this problem in an appropriate space V and characterize V (2) Show that the corresponding bilinear form is coercive and continuous in V and the linear form is continuous in V (3) Set up a finite dimensional space V h V of piece-wise polynomial functions over a uniform partition of (0, 1) Define the nodal basis in terms of the degrees of freedom (4) Introduce the Galerkin method for the problem (??) for V h ; state the error estimate in the V -norm assuming smooth solution u(x) Problem 5: Consider the following boundary value problem for the biharmonic equation: find u(x) that satisfies the differential equation (5) u u u = f(x), x,
8 8 RAYTCHO LAZAROV and homogeneous Dirichlet conditions on the boundary Γ, (6) u(x) = 0 and u(x) n = 0, x Γ Here is the unit square in R 2, Γ is its boundary, and f(x) L 2 () is given Here 1 u denotes the derivative x1 u and 2 u denotes the derivative x2 u The problem (5)-(6) describes the transverse deformation of homogeneous isotropic plate with constant thickness and clamped boundary due to external force f(x) (1) Derive a weak formulation of this problem which gives rise to a symmetric bilinear form on V = H 2 0 () {φ H2 1 () : φ = 1φ = 2 φ = 0 on Γ} (2) Show that the bilinear from is coercive on V (Hint: you may use the Poincaré inequality on H 1 0 ()) (3) Let T h be a triangulation of into triangles Consider the FE (K, P K, Σ), where: (α) K T h is a triangle determined by its three vertexes z 1, z 2, z 3, (β) P K = P 5 (K) is the set of polynomials of degree 5 over K, and (γ) Σ = {v(z i ), 1 v(z i ), 2 v(z i ), 2 11 v(z i), 2 12 v(z i), 2 22 v(z i), ν v(m i ), i = 1, 2, 3}, with m i denoting the midpoints of the triangle sides Show that the finite dimensional space V h based on the finite element (K, P K, Σ) is a subspace of H 2 () Problem 6: Let = (0, 1) and u be the solution of the boundary value problem u (4) (k(x)u ) + c(x)u = f(x) x, u(0) = u (0) = 0, u(1) = 0, u (1) + βu (1) = γ, where k(x) 0, c(x) 0, f(x), γ, and β > 0 are given data Derive the weak formulation of this problem Specify the appropriate Sobolev spaces and show that the corresponding bilinear form is coercive Problem 7: Consider the variational problem: find u H 1 (), such that a(u, v) = l(v) for all v H 1 (), where = (0, 1) (0, 1), Γ is its boundary, and 1 a(u, v) = u v dx + u(x, 0)v(x, 0) dx and l(v) = gvds (a) Derive the strong form to this problem (b) Prove that the bilinear form a(u, v) is coercive in H 1 () Problem 8: Derive weak formulation of the Stokes system 0 References [1] L C Evans, Partial Differential Equations, Graduate Studies in Mathematics, vol 19, AMS, 1998 [2] Ch Grossmann, H-O Ross, and M Stynes, Numerical Treatment of Partial Differential Equations, Springer, Berlin, 2005 [3] M Renardy and R Rogers, An Introduction to Partial Differential Equations, Texts in Applied Mathematics, Springer-Verlag, 1993 [4] P Knabner and L Angermann, Numerical Methods for Elliptic and Parabolic PDEs, Springer-Verlag, New Yrok Inc, 2003 [5] J Wloka, Partial Differential Equations, Cambridge University Press, 1992 Γ
INTRODUCTION TO PDEs
INTRODUCTION TO PDEs In this course we are interested in the numerical approximation of PDEs using finite difference methods (FDM). We will use some simple prototype boundary value problems (BVP) and initial
More informationSecond Order Elliptic PDE
Second Order Elliptic PDE T. Muthukumar tmk@iitk.ac.in December 16, 2014 Contents 1 A Quick Introduction to PDE 1 2 Classification of Second Order PDE 3 3 Linear Second Order Elliptic Operators 4 4 Periodic
More informationScientific Computing WS 2018/2019. Lecture 15. Jürgen Fuhrmann Lecture 15 Slide 1
Scientific Computing WS 2018/2019 Lecture 15 Jürgen Fuhrmann juergen.fuhrmann@wias-berlin.de Lecture 15 Slide 1 Lecture 15 Slide 2 Problems with strong formulation Writing the PDE with divergence and gradient
More informationTheory of PDE Homework 2
Theory of PDE Homework 2 Adrienne Sands April 18, 2017 In the following exercises we assume the coefficients of the various PDE are smooth and satisfy the uniform ellipticity condition. R n is always an
More informationLecture Notes on PDEs
Lecture Notes on PDEs Alberto Bressan February 26, 2012 1 Elliptic equations Let IR n be a bounded open set Given measurable functions a ij, b i, c : IR, consider the linear, second order differential
More informationNumerical Solutions to Partial Differential Equations
Numerical Solutions to Partial Differential Equations Zhiping Li LMAM and School of Mathematical Sciences Peking University Sobolev Embedding Theorems Embedding Operators and the Sobolev Embedding Theorem
More informationFinite Elements. Colin Cotter. February 22, Colin Cotter FEM
Finite Elements February 22, 2019 In the previous sections, we introduced the concept of finite element spaces, which contain certain functions defined on a domain. Finite element spaces are examples of
More informationTraces and Duality Lemma
Traces and Duality Lemma Recall the duality lemma with H / ( ) := γ 0 (H ()) defined as the trace space of H () endowed with minimal extension norm; i.e., for w H / ( ) L ( ), w H / ( ) = min{ ŵ H () ŵ
More informationNumerical Analysis and Methods for PDE I
Numerical Analysis and Methods for PDE I A. J. Meir Department of Mathematics and Statistics Auburn University US-Africa Advanced Study Institute on Analysis, Dynamical Systems, and Mathematical Modeling
More informationLecture Note III: Least-Squares Method
Lecture Note III: Least-Squares Method Zhiqiang Cai October 4, 004 In this chapter, we shall present least-squares methods for second-order scalar partial differential equations, elastic equations of solids,
More informationCLASSIFICATION AND PRINCIPLE OF SUPERPOSITION FOR SECOND ORDER LINEAR PDE
CLASSIFICATION AND PRINCIPLE OF SUPERPOSITION FOR SECOND ORDER LINEAR PDE 1. Linear Partial Differential Equations A partial differential equation (PDE) is an equation, for an unknown function u, that
More informationPDEs, part 1: Introduction and elliptic PDEs
PDEs, part 1: Introduction and elliptic PDEs Anna-Karin Tornberg Mathematical Models, Analysis and Simulation Fall semester, 2013 Partial di erential equations The solution depends on several variables,
More informationChapter 12. Partial di erential equations Di erential operators in R n. The gradient and Jacobian. Divergence and rotation
Chapter 12 Partial di erential equations 12.1 Di erential operators in R n The gradient and Jacobian We recall the definition of the gradient of a scalar function f : R n! R, as @f grad f = rf =,..., @f
More informationLECTURE 1: SOURCES OF ERRORS MATHEMATICAL TOOLS A PRIORI ERROR ESTIMATES. Sergey Korotov,
LECTURE 1: SOURCES OF ERRORS MATHEMATICAL TOOLS A PRIORI ERROR ESTIMATES Sergey Korotov, Institute of Mathematics Helsinki University of Technology, Finland Academy of Finland 1 Main Problem in Mathematical
More informationFinite Elements. Colin Cotter. January 15, Colin Cotter FEM
Finite Elements January 15, 2018 Why Can solve PDEs on complicated domains. Have flexibility to increase order of accuracy and match the numerics to the physics. has an elegant mathematical formulation
More informationSimple Examples on Rectangular Domains
84 Chapter 5 Simple Examples on Rectangular Domains In this chapter we consider simple elliptic boundary value problems in rectangular domains in R 2 or R 3 ; our prototype example is the Poisson equation
More informationNumerical Solutions to Partial Differential Equations
Numerical Solutions to Partial Differential Equations Zhiping Li LMAM and School of Mathematical Sciences Peking University Numerical Methods for Partial Differential Equations Finite Difference Methods
More informationTHE STOKES SYSTEM R.E. SHOWALTER
THE STOKES SYSTEM R.E. SHOWALTER Contents 1. Stokes System 1 Stokes System 2 2. The Weak Solution of the Stokes System 3 3. The Strong Solution 4 4. The Normal Trace 6 5. The Mixed Problem 7 6. The Navier-Stokes
More informationM.Sc. in Meteorology. Numerical Weather Prediction
M.Sc. in Meteorology UCD Numerical Weather Prediction Prof Peter Lynch Meteorology & Climate Centre School of Mathematical Sciences University College Dublin Second Semester, 2005 2006. In this section
More informationIntroduction. J.M. Burgers Center Graduate Course CFD I January Least-Squares Spectral Element Methods
Introduction In this workshop we will introduce you to the least-squares spectral element method. As you can see from the lecture notes, this method is a combination of the weak formulation derived from
More informationSolutions of Selected Problems
1 Solutions of Selected Problems October 16, 2015 Chapter I 1.9 Consider the potential equation in the disk := {(x, y) R 2 ; x 2 +y 2 < 1}, with the boundary condition u(x) = g(x) r for x on the derivative
More informationINTRODUCTION TO FINITE ELEMENT METHODS
INTRODUCTION TO FINITE ELEMENT METHODS LONG CHEN Finite element methods are based on the variational formulation of partial differential equations which only need to compute the gradient of a function.
More informationLecture 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[2] (a) Develop and describe the piecewise linear Galerkin finite element approximation of,
269 C, Vese Practice problems [1] Write the differential equation u + u = f(x, y), (x, y) Ω u = 1 (x, y) Ω 1 n + u = x (x, y) Ω 2, Ω = {(x, y) x 2 + y 2 < 1}, Ω 1 = {(x, y) x 2 + y 2 = 1, x 0}, Ω 2 = {(x,
More informationVariational Formulations
Chapter 2 Variational Formulations In this chapter we will derive a variational (or weak) formulation of the elliptic boundary value problem (1.4). We will discuss all fundamental theoretical results that
More informationA very short introduction to the Finite Element Method
A very short introduction to the Finite Element Method Till Mathis Wagner Technical University of Munich JASS 2004, St Petersburg May 4, 2004 1 Introduction This is a short introduction to the finite element
More informationAM 205: lecture 14. Last time: Boundary value problems Today: Numerical solution of PDEs
AM 205: lecture 14 Last time: Boundary value problems Today: Numerical solution of PDEs ODE BVPs A more general approach is to formulate a coupled system of equations for the BVP based on a finite difference
More informationFinite difference method for elliptic problems: I
Finite difference method for elliptic problems: I Praveen. C praveen@math.tifrbng.res.in Tata Institute of Fundamental Research Center for Applicable Mathematics Bangalore 560065 http://math.tifrbng.res.in/~praveen
More information1. Let a(x) > 0, and assume that u and u h are the solutions of the Dirichlet problem:
Mathematics Chalmers & GU TMA37/MMG800: Partial Differential Equations, 011 08 4; kl 8.30-13.30. Telephone: Ida Säfström: 0703-088304 Calculators, formula notes and other subject related material are not
More informationCONVERGENCE THEORY. G. ALLAIRE CMAP, Ecole Polytechnique. 1. Maximum principle. 2. Oscillating test function. 3. Two-scale convergence
1 CONVERGENCE THEOR G. ALLAIRE CMAP, Ecole Polytechnique 1. Maximum principle 2. Oscillating test function 3. Two-scale convergence 4. Application to homogenization 5. General theory H-convergence) 6.
More information3 2 6 Solve the initial value problem u ( t) 3. a- If A has eigenvalues λ =, λ = 1 and corresponding eigenvectors 1
Math Problem a- If A has eigenvalues λ =, λ = 1 and corresponding eigenvectors 1 3 6 Solve the initial value problem u ( t) = Au( t) with u (0) =. 3 1 u 1 =, u 1 3 = b- True or false and why 1. if A is
More informationRegularity Theory a Fourth Order PDE with Delta Right Hand Side
Regularity Theory a Fourth Order PDE with Delta Right Hand Side Graham Hobbs Applied PDEs Seminar, 29th October 2013 Contents Problem and Weak Formulation Example - The Biharmonic Problem Regularity Theory
More informationPartial Differential Equations
Part II Partial Differential Equations Year 2015 2014 2013 2012 2011 2010 2009 2008 2007 2006 2005 2015 Paper 4, Section II 29E Partial Differential Equations 72 (a) Show that the Cauchy problem for u(x,
More informationBasic Principles of Weak Galerkin Finite Element Methods for PDEs
Basic Principles of Weak Galerkin Finite Element Methods for PDEs Junping Wang Computational Mathematics Division of Mathematical Sciences National Science Foundation Arlington, VA 22230 Polytopal Element
More informationNumerical Solutions to Partial Differential Equations
Numerical Solutions to Partial Differential Equations Zhiping Li LMAM and School of Mathematical Sciences Peking University Variational Problems of the Dirichlet BVP of the Poisson Equation 1 For the homogeneous
More informationChapter 2 Boundary and Initial Data
Chapter 2 Boundary and Initial Data Abstract This chapter introduces the notions of boundary and initial value problems. Some operator notation is developed in order to represent boundary and initial value
More informationDiscontinuous Galerkin Methods
Discontinuous Galerkin Methods Joachim Schöberl May 20, 206 Discontinuous Galerkin (DG) methods approximate the solution with piecewise functions (polynomials), which are discontinuous across element interfaces.
More informationChapter 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 informationChapter 1 Foundations of Elliptic Boundary Value Problems 1.1 Euler equations of variational problems
Chapter 1 Foundations of Elliptic Boundary Value Problems 1.1 Euler equations of variational problems Elliptic boundary value problems often occur as the Euler equations of variational problems the latter
More informationMixed Finite Element Methods. Douglas N. Arnold, University of Minnesota The 41st Woudschoten Conference 5 October 2016
Mixed Finite Element Methods Douglas N. Arnold, University of Minnesota The 41st Woudschoten Conference 5 October 2016 Linear elasticity Given the load f : Ω R n, find the displacement u : Ω R n and the
More informationTable of Contents. II. PDE classification II.1. Motivation and Examples. II.2. Classification. II.3. Well-posedness according to Hadamard
Table of Contents II. PDE classification II.. Motivation and Examples II.2. Classification II.3. Well-posedness according to Hadamard Chapter II (ContentChapterII) Crashtest: Reality Simulation http:www.ara.comprojectssvocrownvic.htm
More informationIntroduction to Partial Differential Equations
Introduction to Partial Differential Equations Philippe B. Laval KSU Current Semester Philippe B. Laval (KSU) Key Concepts Current Semester 1 / 25 Introduction The purpose of this section is to define
More informationScientific Computing WS 2017/2018. Lecture 18. Jürgen Fuhrmann Lecture 18 Slide 1
Scientific Computing WS 2017/2018 Lecture 18 Jürgen Fuhrmann juergen.fuhrmann@wias-berlin.de Lecture 18 Slide 1 Lecture 18 Slide 2 Weak formulation of homogeneous Dirichlet problem Search u H0 1 (Ω) (here,
More informationAppendix A Functional Analysis
Appendix A Functional Analysis A.1 Metric Spaces, Banach Spaces, and Hilbert Spaces Definition A.1. Metric space. Let X be a set. A map d : X X R is called metric on X if for all x,y,z X it is i) d(x,y)
More informationApplied/Numerical Analysis Qualifying Exam
Applied/Numerical Analysis Qualifying Exam August 9, 212 Cover Sheet Applied Analysis Part Policy on misprints: The qualifying exam committee tries to proofread exams as carefully as possible. Nevertheless,
More informationarxiv: v1 [math.na] 29 Feb 2016
EFFECTIVE IMPLEMENTATION OF THE WEAK GALERKIN FINITE ELEMENT METHODS FOR THE BIHARMONIC EQUATION LIN MU, JUNPING WANG, AND XIU YE Abstract. arxiv:1602.08817v1 [math.na] 29 Feb 2016 The weak Galerkin (WG)
More informationWeak Convergence Methods for Energy Minimization
Weak Convergence Methods for Energy Minimization Bo Li Department of Mathematics University of California, San Diego E-mail: bli@math.ucsd.edu June 3, 2007 Introduction This compact set of notes present
More informationVariational Principles for Equilibrium Physical Systems
Variational Principles for Equilibrium Physical Systems 1. Variational Principles One way of deriving the governing equations for a physical system is the express the relevant conservation statements and
More informationApplied Math Qualifying Exam 11 October Instructions: Work 2 out of 3 problems in each of the 3 parts for a total of 6 problems.
Printed Name: Signature: Applied Math Qualifying Exam 11 October 2014 Instructions: Work 2 out of 3 problems in each of the 3 parts for a total of 6 problems. 2 Part 1 (1) Let Ω be an open subset of R
More informationt y n (s) ds. t y(s) ds, x(t) = x(0) +
1 Appendix Definition (Closed Linear Operator) (1) The graph G(T ) of a linear operator T on the domain D(T ) X into Y is the set (x, T x) : x D(T )} in the product space X Y. Then T is closed if its graph
More informationQuestion 9: PDEs Given the function f(x, y), consider the problem: = f(x, y) 2 y2 for 0 < x < 1 and 0 < x < 1. x 2 u. u(x, 0) = u(x, 1) = 0 for 0 x 1
Question 9: PDEs Given the function f(x, y), consider the problem: 2 u x 2 u = f(x, y) 2 y2 for 0 < x < 1 and 0 < x < 1 u(x, 0) = u(x, 1) = 0 for 0 x 1 u(0, y) = u(1, y) = 0 for 0 y 1. a. Discuss how you
More informationTraces, extensions and co-normal derivatives for elliptic systems on Lipschitz domains
Traces, extensions and co-normal derivatives for elliptic systems on Lipschitz domains Sergey E. Mikhailov Brunel University West London, Department of Mathematics, Uxbridge, UB8 3PH, UK J. Math. Analysis
More informationMath The Laplacian. 1 Green s Identities, Fundamental Solution
Math. 209 The Laplacian Green s Identities, Fundamental Solution Let be a bounded open set in R n, n 2, with smooth boundary. The fact that the boundary is smooth means that at each point x the external
More informationFundamental Solutions and Green s functions. Simulation Methods in Acoustics
Fundamental Solutions and Green s functions Simulation Methods in Acoustics Definitions Fundamental solution The solution F (x, x 0 ) of the linear PDE L {F (x, x 0 )} = δ(x x 0 ) x R d Is called the fundamental
More informationLecture 1. Finite difference and finite element methods. Partial differential equations (PDEs) Solving the heat equation numerically
Finite difference and finite element methods Lecture 1 Scope of the course Analysis and implementation of numerical methods for pricing options. Models: Black-Scholes, stochastic volatility, exponential
More informationA metric space X is a non-empty set endowed with a metric ρ (x, y):
Chapter 1 Preliminaries References: Troianiello, G.M., 1987, Elliptic differential equations and obstacle problems, Plenum Press, New York. Friedman, A., 1982, Variational principles and free-boundary
More informationLecture Notes: African Institute of Mathematics Senegal, January Topic Title: A short introduction to numerical methods for elliptic PDEs
Lecture Notes: African Institute of Mathematics Senegal, January 26 opic itle: A short introduction to numerical methods for elliptic PDEs Authors and Lecturers: Gerard Awanou (University of Illinois-Chicago)
More informationGreen s Functions and Distributions
CHAPTER 9 Green s Functions and Distributions 9.1. Boundary Value Problems We would like to study, and solve if possible, boundary value problems such as the following: (1.1) u = f in U u = g on U, where
More informationFinite Element Methods for Fourth Order Variational Inequalities
Louisiana State University LSU Digital Commons LSU Doctoral Dissertations Graduate School 2013 Finite Element Methods for Fourth Order Variational Inequalities Yi Zhang Louisiana State University and Agricultural
More informationThe Dirichlet s P rinciple. In this lecture we discuss an alternative formulation of the Dirichlet problem for the Laplace equation:
Oct. 1 The Dirichlet s P rinciple In this lecture we discuss an alternative formulation of the Dirichlet problem for the Laplace equation: 1. Dirichlet s Principle. u = in, u = g on. ( 1 ) If we multiply
More information1 The Stokes System. ρ + (ρv) = ρ g(x), and the conservation of momentum has the form. ρ v (λ 1 + µ 1 ) ( v) µ 1 v + p = ρ f(x) in Ω.
1 The Stokes System The motion of a (possibly compressible) homogeneous fluid is described by its density ρ(x, t), pressure p(x, t) and velocity v(x, t). Assume that the fluid is barotropic, i.e., the
More informationAdaptive Finite Element Methods Lecture Notes Winter Term 2017/18. R. Verfürth. Fakultät für Mathematik, Ruhr-Universität Bochum
Adaptive Finite Element Methods Lecture Notes Winter Term 2017/18 R. Verfürth Fakultät für Mathematik, Ruhr-Universität Bochum Contents Chapter I. Introduction 7 I.1. Motivation 7 I.2. Sobolev and finite
More informationFinite Elements for Elastic Shell Models in
Elastic s in Advisor: Matthias Heinkenschloss Computational and Applied Mathematics Rice University 13 April 2007 Outline Elasticity in Differential Geometry of Shell Geometry and Equations The Plate Model
More informationMath Tune-Up Louisiana State University August, Lectures on Partial Differential Equations and Hilbert Space
Math Tune-Up Louisiana State University August, 2008 Lectures on Partial Differential Equations and Hilbert Space 1. A linear partial differential equation of physics We begin by considering the simplest
More information1. Introduction. We consider the model problem that seeks an unknown function u = u(x) satisfying
A SIMPLE FINITE ELEMENT METHOD FOR LINEAR HYPERBOLIC PROBLEMS LIN MU AND XIU YE Abstract. In this paper, we introduce a simple finite element method for solving first order hyperbolic equations with easy
More informationModule 7: The Laplace Equation
Module 7: The Laplace Equation In this module, we shall study one of the most important partial differential equations in physics known as the Laplace equation 2 u = 0 in Ω R n, (1) where 2 u := n i=1
More informationPartial Differential Equations
M3M3 Partial Differential Equations Solutions to problem sheet 3/4 1* (i) Show that the second order linear differential operators L and M, defined in some domain Ω R n, and given by Mφ = Lφ = j=1 j=1
More informationFrom Completing the Squares and Orthogonal Projection to Finite Element Methods
From Completing the Squares and Orthogonal Projection to Finite Element Methods Mo MU Background In scientific computing, it is important to start with an appropriate model in order to design effective
More information1 Discretizing BVP with Finite Element Methods.
1 Discretizing BVP with Finite Element Methods In this section, we will discuss a process for solving boundary value problems numerically, the Finite Element Method (FEM) We note that such method is a
More informationLucio Demeio Dipartimento di Ingegneria Industriale e delle Scienze Matematiche
Scuola di Dottorato THE WAVE EQUATION Lucio Demeio Dipartimento di Ingegneria Industriale e delle Scienze Matematiche Lucio Demeio - DIISM wave equation 1 / 44 1 The Vibrating String Equation 2 Second
More informationFact Sheet Functional Analysis
Fact Sheet Functional Analysis Literature: Hackbusch, W.: Theorie und Numerik elliptischer Differentialgleichungen. Teubner, 986. Knabner, P., Angermann, L.: Numerik partieller Differentialgleichungen.
More informationNumerical Methods for the Navier-Stokes equations
Arnold Reusken Numerical Methods for the Navier-Stokes equations January 6, 212 Chair for Numerical Mathematics RWTH Aachen Contents 1 The Navier-Stokes equations.............................................
More informationIndex. C 2 ( ), 447 C k [a,b], 37 C0 ( ), 618 ( ), 447 CD 2 CN 2
Index advection equation, 29 in three dimensions, 446 advection-diffusion equation, 31 aluminum, 200 angle between two vectors, 58 area integral, 439 automatic step control, 119 back substitution, 604
More informationWeak Galerkin Finite Element Scheme and Its Applications
Weak Galerkin Finite Element Scheme and Its Applications Ran Zhang Department of Mathematics Jilin University, China IMS, Singapore February 6, 2015 Talk Outline Motivation WG FEMs: Weak Operators + Stabilizer
More informationCIV-E1060 Engineering Computation and Simulation Examination, December 12, 2017 / Niiranen
CIV-E16 Engineering Computation and Simulation Examination, December 12, 217 / Niiranen This examination consists of 3 problems rated by the standard scale 1...6. Problem 1 Let us consider a long and tall
More informationand finally, any second order divergence form elliptic operator
Supporting Information: Mathematical proofs Preliminaries Let be an arbitrary bounded open set in R n and let L be any elliptic differential operator associated to a symmetric positive bilinear form B
More informationAMS subject classifications. Primary, 65N15, 65N30, 76D07; Secondary, 35B45, 35J50
A SIMPLE FINITE ELEMENT METHOD FOR THE STOKES EQUATIONS LIN MU AND XIU YE Abstract. The goal of this paper is to introduce a simple finite element method to solve the Stokes equations. This method is in
More informationWELL POSEDNESS OF PROBLEMS I
Finite Element Method 85 WELL POSEDNESS OF PROBLEMS I Consider the following generic problem Lu = f, where L : X Y, u X, f Y and X, Y are two Banach spaces We say that the above problem is well-posed (according
More informationGeneralized Lax-Milgram theorem in Banach spaces and its application to the mathematical fluid mechanics.
Second Italian-Japanese Workshop GEOMETRIC PROPERTIES FOR PARABOLIC AND ELLIPTIC PDE s Cortona, Italy, June 2-24, 211 Generalized Lax-Milgram theorem in Banach spaces and its application to the mathematical
More informationNumerical Analysis of Differential Equations Numerical Solution of Elliptic Boundary Value
Numerical Analysis of Differential Equations 188 5 Numerical Solution of Elliptic Boundary Value Problems 5 Numerical Solution of Elliptic Boundary Value Problems TU Bergakademie Freiberg, SS 2012 Numerical
More informationIMPROVED LEAST-SQUARES ERROR ESTIMATES FOR SCALAR HYPERBOLIC PROBLEMS, 1
Computational Methods in Applied Mathematics Vol. 1, No. 1(2001) 1 8 c Institute of Mathematics IMPROVED LEAST-SQUARES ERROR ESTIMATES FOR SCALAR HYPERBOLIC PROBLEMS, 1 P.B. BOCHEV E-mail: bochev@uta.edu
More informationarxiv: v1 [math.na] 27 Jan 2016
Virtual Element Method for fourth order problems: L 2 estimates Claudia Chinosi a, L. Donatella Marini b arxiv:1601.07484v1 [math.na] 27 Jan 2016 a Dipartimento di Scienze e Innovazione Tecnologica, Università
More informationi=1 α i. Given an m-times continuously
1 Fundamentals 1.1 Classification and characteristics Let Ω R d, d N, d 2, be an open set and α = (α 1,, α d ) T N d 0, N 0 := N {0}, a multiindex with α := d i=1 α i. Given an m-times continuously differentiable
More informationWeak Solutions, Elliptic Problems and Sobolev Spaces
3 Weak Solutions, Elliptic Problems and Sobolev Spaces 3.1 Introduction In Chapter 2 we discussed difference methods for the numerical treatment of partial differential equations. The basic idea of these
More informationAMS 529: Finite Element Methods: Fundamentals, Applications, and New Trends
AMS 529: Finite Element Methods: Fundamentals, Applications, and New Trends Lecture 3: Finite Elements in 2-D Xiangmin Jiao SUNY Stony Brook Xiangmin Jiao Finite Element Methods 1 / 18 Outline 1 Boundary
More informationThe Navier-Stokes problem in velocity-pressure formulation :convergence and Optimal Control
The Navier-Stokes problem in velocity-pressure formulation :convergence and Optimal Control A.Younes 1 A. Jarray 2 1 Faculté des Sciences de Tunis, Tunisie. e-mail :younesanis@yahoo.fr 2 Faculté des Sciences
More informationAn introduction to the mathematical theory of finite elements
Master in Seismic Engineering E.T.S.I. Industriales (U.P.M.) Discretization Methods in Engineering An introduction to the mathematical theory of finite elements Ignacio Romero ignacio.romero@upm.es October
More informationThe continuity method
The continuity method The method of continuity is used in conjunction with a priori estimates to prove the existence of suitably regular solutions to elliptic partial differential equations. One crucial
More informationPARTIAL DIFFERENTIAL EQUATIONS. Lecturer: D.M.A. Stuart MT 2007
PARTIAL DIFFERENTIAL EQUATIONS Lecturer: D.M.A. Stuart MT 2007 In addition to the sets of lecture notes written by previous lecturers ([1, 2]) the books [4, 7] are very good for the PDE topics in the course.
More informationDiscretization of PDEs and Tools for the Parallel Solution of the Resulting Systems
Discretization of PDEs and Tools for the Parallel Solution of the Resulting Systems Stan Tomov Innovative Computing Laboratory Computer Science Department The University of Tennessee Wednesday April 4,
More informationPREPRINT 2010:23. A nonconforming rotated Q 1 approximation on tetrahedra PETER HANSBO
PREPRINT 2010:23 A nonconforming rotated Q 1 approximation on tetrahedra PETER HANSBO Department of Mathematical Sciences Division of Mathematics CHALMERS UNIVERSITY OF TECHNOLOGY UNIVERSITY OF GOTHENBURG
More informationMath 660-Lecture 15: Finite element spaces (I)
Math 660-Lecture 15: Finite element spaces (I) (Chapter 3, 4.2, 4.3) Before we introduce the concrete spaces, let s first of all introduce the following important lemma. Theorem 1. Let V h consists of
More informationChapter 1: The Finite Element Method
Chapter 1: The Finite Element Method Michael Hanke Read: Strang, p 428 436 A Model Problem Mathematical Models, Analysis and Simulation, Part Applications: u = fx), < x < 1 u) = u1) = D) axial deformation
More informationPDEs in Image Processing, Tutorials
PDEs in Image Processing, Tutorials Markus Grasmair Vienna, Winter Term 2010 2011 Direct Methods Let X be a topological space and R: X R {+ } some functional. following definitions: The mapping R is lower
More informationMIXED FINITE ELEMENTS FOR PLATES. Ricardo G. Durán Universidad de Buenos Aires
MIXED FINITE ELEMENTS FOR PLATES Ricardo G. Durán Universidad de Buenos Aires - Necessity of 2D models. - Reissner-Mindlin Equations. - Finite Element Approximations. - Locking. - Mixed interpolation or
More informationSome lecture notes for Math 6050E: PDEs, Fall 2016
Some lecture notes for Math 65E: PDEs, Fall 216 Tianling Jin December 1, 216 1 Variational methods We discuss an example of the use of variational methods in obtaining existence of solutions. Theorem 1.1.
More informationNumerical Methods for Partial Differential Equations: an Overview.
Numerical Methods for Partial Differential Equations: an Overview math652_spring2009@colorstate PDEs are mathematical models of physical phenomena Heat conduction Wave motion PDEs are mathematical models
More informationSHARP BOUNDARY TRACE INEQUALITIES. 1. Introduction
SHARP BOUNDARY TRACE INEQUALITIES GILES AUCHMUTY Abstract. This paper describes sharp inequalities for the trace of Sobolev functions on the boundary of a bounded region R N. The inequalities bound (semi-)norms
More informationPartial Differential Equations 2 Variational Methods
Partial Differential Equations 2 Variational Methods Martin Brokate Contents 1 Variational Methods: Some Basics 1 2 Sobolev Spaces: Definition 9 3 Elliptic Boundary Value Problems 2 4 Boundary Conditions,
More informationIntroduction of Partial Differential Equations and Boundary Value Problems
Introduction of Partial Differential Equations and Boundary Value Problems 2009 Outline Definition Classification Where PDEs come from? Well-posed problem, solutions Initial Conditions and Boundary Conditions
More information