TMA4220 Numerical Solution of Partial Differential Equations Using Element Methods Høst 2014
|
|
- Alan Perry
- 5 years ago
- Views:
Transcription
1 Norwegian University of Science and Technology Department of Mathematical Sciences TMA422 Numerical Solution of Partial Differential Equations Using Element Methods Høst 214 Exercise set 2 1 Consider the equation u xx = f(x), < x < 1, u() =, u(1) =, which is solved with the finite element method with equidistant grid (x i = ih, h = 1/N), and the basis functions x x i 1 h, for x i 1 x x i, x ϕ i (x) = i+1 x h for x i x x i+1, otherwise. for i = 1,2,...,N 1. Denote the numerical solution u N. In Exercise 1.3, you solved this equation with f(x) = x 4. You may have noticed that the numerical and exact solution coincide on the gridpoints, that is u(x i ) = u N (x i ), for i =,1,2,...,N. In this problem, we are to show that this is true for all f(x). For i = and i = N, there is nothing to prove (Why?), what remains is i = 1,2,...,N 1. a) Prove that a(u u h,ϕ i ) = for i = 1,2,...,N 1. Solution: Let V h = span{ϕ 1,ϕ 2,...,ϕ N 1 } V = H 1 ([,1]). Then the numerical solution satisfies a(u h,v h ) = f(v h ) v V h, while the exact solution satisfies ϕ i V h V, so a(u,v) = f(v) v V. a(u u h,ϕ i ) = a(u,ϕ i ) a(u h,ϕ i ) a bilinear. = f(ϕ i ) f(ϕ i ) u and u h solves the wea problems. =. b) Prove that for all v H 1 ([,1]) C ([,1]) and i = 1,2...,N 1, a(v,ϕ i ) = 1 h (2v(x i) v(x i 1 ) v(x i+1 )). September 1, 214 Page 1 of 7
2 Solution: a(v,ϕ i ) = v x ϕ i. 1 ϕ h, for x i 1 < x < x i, i = 1 h, for x i < x < x i+1,, otherwise. a(v,ϕ i ) = xi x i 1 v x h xi+1 x i v x h c) The M M matrix = 1 h (v(x i) v(x i 1 ) (v(x i+1 ) v(x i ))) = 1 h (2v(x i) v(x i 1 ) v(x i+1 )) A = is invertible for all M. Use this fact and the results of a) and b) to prove that u(x i ) u N (x i ) = for i = 1,2...,N 1. Solution: Combining the results of a) and b), we see that v i = u(x i ) u N (x i ) satisfies v i 1 +2v i v i+1 = for i = 1,2,...,N 1. Removing v and v N, (which are equal to zero) this is equivalent to Av = where A is as above and v = [v 1 v 2...v N 1 ]. Since A is invertible, Av = v =. 2 Given the Helmholtz problem u xx +σu = f on (,1), u() = u(1) =. where σ > is a constant. Set up the wea form for this problem. Show that, when this problem is solved by a Galerin method, using V h = span {φ i } N i=1, the discrete problem can be written as (A+σM)u = f. Set up the matrix M for V h = Xh 1 on a uniform grid. Solution: Multiply by the equation by a test function v, integrate over the domain September 1, 214 Page 2 of 7
3 (,1), use partial integration and get rid of the boundary terms by require v() = v(1) =. The the wea formualtion becomes: Find u H 1 (Ω) such that v x u x +σ uv = fv, v H 1 (,1). Choose V h = span{ϕ 1,ϕ 2,...,ϕ N }, let our unnown approximation be written as u h (x) = where the coefficients u j is found from u j ϕ j (x), ( dϕj ) dϕ i +σ (ϕ j ϕ i ) = (f ϕ i ), i = 1,2,,N. This is a linear system of equations Au+σMu = f where (A i,j = ( dϕj ) dϕ i, M i,j = (ϕ j ϕ i ) and f i = (fϕ i ). Let V h = X 1 h on a uniform grid, so that h = 1/N and x i = ih, i =,1,...,N. ϕ i 1 ϕ i ϕ i+1 h h h h x i 2 x i 1 x i x i+1 x i+2 The non-zeros elements of the stiffness matrix A and the mass matrix M is xi+1 ( ) dϕi A i,i = = 2 xi+1 h, A dϕ i i,i+1 = = 1 h = A i+1,i M i,i = x i 1 xi+1 x i 1 ϕ 2 i = 2 3 h, M i,i+1 = dϕ i+1 x i xi+1 x i ϕ i+1 ϕ i = 1 6 h = M i+1,i. All the other elements are. So 4 1 M = h R (N 1) (N 1). September 1, 214 Page 3 of 7
4 3 Write a MATLAB program for solving the Helmholtz problem u xx +σu = f(x), < x < 1, u() = u(1) =. or, using the wea formulation find u H 1 (,1) s.t. u x v x +σ uv = fv, for all v H 1 (,1) (1) by the finite element method on Xh 2, using the algorithm outlined in the supplementary note. To test your code, let σ = 1, f = sin(πx) in which case u(x) = sin(πx)/(1+π 2 ). Use for example [,.1,.25,.3,.4,.45,.5,.55,.6,.7,.8,.9,1] for the partition of the elements. As already pointed out in exercise 2, the discrete problem can be written as (A+σM)u = b. (2) So the tas is to set up the matrices A and M and the load vector b, and solve the system. What you have to do is described in the following: a) Preliminaries: Set up the element matrices A K h and MK h, corresponding to contribution from element K to the first and second integrals of (1) resp. Solution: Starting from the very beginning: The quadratic shape functions defined on the reference element ˆK = (,1), corresponding to the nodes x 1 =, x 2 = 1/2 and x 3 = 1 is given by ψ 1 (ξ) = 2(ξ 1 2 )(ξ 1), ψ 2(x) = 4ξ(ξ 1), ψ 3 (x) = 2ξ(ξ 1 2 ). The mapping from the ˆK to K = (,+1 ) is given by x(ξ) = + h ξ and the inverse mapping is ξ(x) = (x )/h, where h = +1. A K h = , Mh K 3h = h So the contribution from element K to the element matrices Mh K and AK h becomes: (M K h ) α,β = (A K h ) α,β = x+1 x+1 ϕ K α (x)ϕ K β (x) = h dϕ K α ψ α (ξ)ψ β (ξ)dξ dϕ K β = 1 ψ α dψ β h dξ dξ dξ se the slides on the webpage for details. Altogether, we ends up with: A K h = , Mh K 3h = h September 1, 214 Page 4 of 7
5 b) Write a function computing integrals by the following quadrature formula: g(x) 1 2 (g(c 1)+g(c 2 )), c 1,2 = 1 2 ± 3 6. This will be used for to approximate the contribution from an element to the load vector. Solution: On an element, the approximation becomes x+1 g(x) g( +c 1 h )+g( +c 2 h ) 2 But the contributions to the load vector is +1 b K i = x+1 The corresponding function can be ϕ i (x)f(x) φ i (x)f(x), so that 2 ψ i (c j )f( +c j ). function b = b(f,x) % Calculate a numerical approximation to the elemental load vector % of the function f on an element [x(1),x(2)]. % using quadratic elements. h = x(2)-x(1); c1 = 1/2-sqrt(3)/6; c2 = 1/2+sqrt(3)/6; b = h*[(c1-.5)*(c1-1)*f(x(1)+h*c1) + (c2-.5)*(c2-1)*f(x(1)+h*c2);... 2*c1*(1-c1)*f(x(1)+h*c1) + 2*c2*(1-c2)*f(x(1)+h*c2); c1*(c1-.5)*f(x(1)+h*c1)+c2*(c2-.5)*f(x(1)+h*c2)]; c) Assemble the prototype matrices Ãh and M h as well as the load vector b.. Solution: For points b)-e), see Figure 1. d) Remove the rows and columns corresponding to the boundary conditions. e) Solve (2), and plot the solution. f) Change the boundary conditions to u() = 1 and u x (1) = 2. Which changes has to be done in the code? Solution: The wea formulation of the Helmholtz problem is now find u V s.t. u x v x +σ uv = fv R g,x v x +2v(1), for all v V where v = {v H 1 (,1) : v() = } and R g H 1 (,1) is some function satisfying R g () = 1. For the numerical solution, we choose R g = ϕ (x). The correpsonding discrete formulation is u j ϕ j,x ϕ i,x +σ u j ϕ j ϕ i = j= fϕ i ϕ i,x ϕ,x + 2φ i (1) for i = 1,2,,N. In practice, this means that the column corresponding to the left hand side Dirichlet condition is moved to the right hand side. In addition, we add one term in the last equation, to include the contribution from the Neumann boundary conditions. Line in Figure 1 has to be changed to September 1, 214 Page 5 of 7
6 f sin(pi*x); sigma = 1; % The partition of [,1]. x = [,.1,.25,.3,.4,.45,.5,.55,.6,.7,.8,.9,1]; A = [7/3, -8/3, 1/3; % Element stiffness matrix -8/3, 16/3, -8/3; 1/3, -8/3, 7/3]; M = [2/15, 1/15, -1/3; % Element mass matrix 1/15, 8/15, 1/15; -1/3, 1/15, 2/15]; N = length(x)-1; % Number of elements. N = 2*N+1; % Number of nodes (including the boundaries) theta 2*(-1)+alpha; % local-to-global mapping. Ah = sparse(n,n); Mh = sparse(n,n); bh = zeros(n,1); % Stiffness matrix % Mass matrix resp. % Load vector % Assemble process: for = 1:N h = x(+1)-x(); % Size of the element gi = theta(,1):theta(,1)+2; Ah(gi,gi) = Ah(gi,gi) + A/h; Mh(gi,gi) = Mh(gi,gi) + M*h; bh(gi) = bh(gi) + b(f,[x(),x(+1)]); end % For the case u()=u(1)= % Remove contributions concerning the Dirichlet boundaries. A = Ah(2:end-1,2:end-1); M = Mh(2:end-1,2:end-1); b = bh(2:end-1); % Solve the system u = (A+sigma*M)\b; % Include the boundaries. u = [;u;]; % For the plot: Create a x-vector with all the nodes % including the midpoints of each element. xi = x(1); for =1:N xi = [xi;x()+(x(+1)-x())/2; x(+1)]; end % Plot the numerical and the exact solution in the nodes. u_exact = sin(pi*xi)/(1+pi^2); plot(xi,u_exact, r,xi,u, b ) legend( u_exact, u_h ); Figure 1: Code for solving the Helmholtz equation in 1D by a quadratic FEM September 1, 214 Page 6 of 7
7 % For the case u()=1, u_x(1)=2. bh = bh - Ah(:,1)*1; bh(end) = bh(end)+2; A = Ah(2:end,2:end); M = Mh(2:end,2:end); b = bh(2:end); % Solve the system u = (A+sigma*M)\b; % Include the Dirichlet boundary. u = [1;u;]; September 1, 214 Page 7 of 7
Lehrstuhl Informatik V. Lehrstuhl Informatik V. 1. solve weak form of PDE to reduce regularity properties. Lehrstuhl Informatik V
Part I: Introduction to Finite Element Methods Scientific Computing I Module 8: An Introduction to Finite Element Methods Tobias Necel Winter 4/5 The Model Problem FEM Main Ingredients Wea Forms and Wea
More informationScientific Computing I
Scientific Computing I Module 8: An Introduction to Finite Element Methods Tobias Neckel Winter 2013/2014 Module 8: An Introduction to Finite Element Methods, Winter 2013/2014 1 Part I: Introduction to
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 informationA brief introduction to finite element methods
CHAPTER A brief introduction to finite element methods 1. Two-point boundary value problem and the variational formulation 1.1. The model problem. Consider the two-point boundary value problem: Given a
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 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 informationTMA4220: Programming project - part 1
TMA4220: Programming project - part 1 TMA4220 - Numerical solution of partial differential equations using the finite element method The programming project will be split into two parts. This is the first
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 informationWRT in 2D: Poisson Example
WRT in 2D: Poisson Example Consider 2 u f on [, L x [, L y with u. WRT: For all v X N, find u X N a(v, u) such that v u dv v f dv. Follows from strong form plus integration by parts: ( ) 2 u v + 2 u dx
More informationThe Finite Element Method for the Wave Equation
The Finite Element Method for the Wave Equation 1 The Wave Equation We consider the scalar wave equation modelling acoustic wave propagation in a bounded domain 3, with boundary Γ : 1 2 u c(x) 2 u 0, in
More information(f P Ω hf) vdx = 0 for all v V h, if and only if Ω (f P hf) φ i dx = 0, for i = 0, 1,..., n, where {φ i } V h is set of hat functions.
Answer ALL FOUR Questions Question 1 Let I = [, 2] and f(x) = (x 1) 3 for x I. (a) Let V h be the continous space of linear piecewise function P 1 (I) on I. Write a MATLAB code to compute the L 2 -projection
More informationFinite Element Method for Ordinary Differential Equations
52 Chapter 4 Finite Element Method for Ordinary Differential Equations In this chapter we consider some simple examples of the finite element method for the approximate solution of ordinary differential
More information1. Differential Equations (ODE and PDE)
1. Differential Equations (ODE and PDE) 1.1. Ordinary Differential Equations (ODE) So far we have dealt with Ordinary Differential Equations (ODE): involve derivatives with respect to only one variable
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 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 informationLecture 26: Putting it all together #5... Galerkin Finite Element Methods for ODE BVPs
Lecture 26: Putting it all together #5... Galerkin Finite Element Methods for ODE BVPs Outline 1) : Review 2) A specific Example: Piecewise linear elements 3) Computational Issues and Algorithms A) Calculation
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 informationTime-dependent variational forms
Time-dependent variational forms Hans Petter Langtangen 1,2 1 Center for Biomedical Computing, Simula Research Laboratory 2 Department of Informatics, University of Oslo Oct 30, 2015 PRELIMINARY VERSION
More informationOne-dimensional and nonlinear problems
Solving PDE s with FEniCS One-dimensional and nonlinear problems L. Ridgway Scott The Institute for Biophysical Dynamics, The Computation Institute, and the Departments of Computer Science and Mathematics,
More informationNumerical Simulation of Potential Flow using the Finite Element Method
Numerical Simulation of Potential Flow using the Finite Element Method Cynthia Vanessa Flores (CSUN), Eduardo Rogério Fávaro(Ibilce-Unesp) Gilberto Pereira Sassi (ICMC-USP) Leonardo Koller Sacht(UFSC)
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 informationDiscretization Methods Exercise # 5
Discretization Methods Exercise # 5 Static calculation of a planar truss structure: a a F Six steps: 1. Discretization 2. Element matrices 3. Transformation 4. Assembly 5. Boundary conditions 6. Solution
More informationLECTURE # 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 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 informationFinite Difference Methods for Boundary Value Problems
Finite Difference Methods for Boundary Value Problems October 2, 2013 () Finite Differences October 2, 2013 1 / 52 Goals Learn steps to approximate BVPs using the Finite Difference Method Start with two-point
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 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 informationImplementation of the dg method
Implementation of the dg method Matteo Cicuttin CERMICS - École Nationale des Ponts et Chaussées Marne-la-Vallée March 6, 2017 Outline Model problem, fix notation Representing polynomials Computing integrals
More informationFinite Element Methods
Solving Operator Equations Via Minimization We start with several definitions. Definition. Let V be an inner product space. A linear operator L: D V V is said to be positive definite if v, Lv > for every
More informationLibMesh Experience and Usage
LibMesh Experience and Usage John W. Peterson peterson@cfdlab.ae.utexas.edu Univ. of Texas at Austin January 12, 2007 1 Introduction 2 Weighted Residuals 3 Poisson Equation 4 Other Examples 5 Essential
More informationFinite Elements for Nonlinear Problems
Finite Elements for Nonlinear Problems Computer Lab 2 In this computer lab we apply finite element method to nonlinear model problems and study two of the most common techniques for solving the resulting
More informationA Finite Element Primer
A Finite Element Primer David J. Silvester Scool of Matematics, University of Mancester d.silvester@mancester.ac.uk. Version.3 updated 4 October Contents A Model Diffusion Problem.................... x.
More informationA Fast Fourier transform based direct solver for the Helmholtz problem. AANMPDE 2017 Palaiochora, Crete, Oct 2, 2017
A Fast Fourier transform based direct solver for the Helmholtz problem Jari Toivanen Monika Wolfmayr AANMPDE 2017 Palaiochora, Crete, Oct 2, 2017 Content 1 Model problem 2 Discretization 3 Fast solver
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 informationScientific Computing: Numerical Integration
Scientific Computing: Numerical Integration Aleksandar Donev Courant Institute, NYU 1 donev@courant.nyu.edu 1 Course MATH-GA.2043 or CSCI-GA.2112, Fall 2015 Nov 5th, 2015 A. Donev (Courant Institute) Lecture
More informationA simple FEM solver and its data parallelism
A simple FEM solver and its data parallelism Gundolf Haase Institute for Mathematics and Scientific Computing University of Graz, Austria Chile, Jan. 2015 Partial differential equation Considered Problem
More informationLibMesh Experience and Usage
LibMesh Experience and Usage John W. Peterson peterson@cfdlab.ae.utexas.edu and Roy H. Stogner roystgnr@cfdlab.ae.utexas.edu Univ. of Texas at Austin September 9, 2008 1 Introduction 2 Weighted Residuals
More informationHigher-Order Compact Finite Element Method
Higher-Order Compact Finite Element Method Major Qualifying Project Advisor: Professor Marcus Sarkis Amorn Chokchaisiripakdee Worcester Polytechnic Institute Abstract The Finite Element Method (FEM) is
More informationPIECEWISE LINEAR FINITE ELEMENT METHODS ARE NOT LOCALIZED
PIECEWISE LINEAR FINITE ELEMENT METHODS ARE NOT LOCALIZED ALAN DEMLOW Abstract. Recent results of Schatz show that standard Galerkin finite element methods employing piecewise polynomial elements of degree
More informationare continuous). Then we also know that the above function g is continuously differentiable on [a, b]. For that purpose, we will first show that
Derivatives and Integrals Now suppose f : [a, b] [c, d] IR is continuous. It is easy to see that for each fixed x [a, b] the function f x (y) = f(x, y) is continuous on [c, d] (note that f x is sometimes
More informationHIGHER-ORDER THEORIES
HIGHER-ORDER THEORIES THIRD-ORDER SHEAR DEFORMATION PLATE THEORY LAYERWISE LAMINATE THEORY J.N. Reddy 1 Third-Order Shear Deformation Plate Theory Assumed Displacement Field µ u(x y z t) u 0 (x y t) +
More informationDomain Decomposition Preconditioners for Spectral Nédélec Elements in Two and Three Dimensions
Domain Decomposition Preconditioners for Spectral Nédélec Elements in Two and Three Dimensions Bernhard Hientzsch Courant Institute of Mathematical Sciences, New York University, 51 Mercer Street, New
More informationyou expect to encounter difficulties when trying to solve A x = b? 4. A composite quadrature rule has error associated with it in the following form
Qualifying exam for numerical analysis (Spring 2017) Show your work for full credit. If you are unable to solve some part, attempt the subsequent parts. 1. Consider the following finite difference: f (0)
More informationIterative Methods for Linear Systems
Iterative Methods for Linear Systems 1. Introduction: Direct solvers versus iterative solvers In many applications we have to solve a linear system Ax = b with A R n n and b R n given. If n is large the
More informationNumerical methods for PDEs FEM convergence, error estimates, piecewise polynomials
Platzhalter für Bild, Bild auf Titelfolie hinter das Logo einsetzen Numerical methods for PDEs FEM convergence, error estimates, piecewise polynomials Dr. Noemi Friedman Contents of the course Fundamentals
More informationClass notes: Approximation
Class notes: Approximation Introduction Vector spaces, linear independence, subspace The goal of Numerical Analysis is to compute approximations We want to approximate eg numbers in R or C vectors in R
More informationChapter 7: Bounded Operators in Hilbert Spaces
Chapter 7: Bounded Operators in Hilbert Spaces I-Liang Chern Department of Applied Mathematics National Chiao Tung University and Department of Mathematics National Taiwan University Fall, 2013 1 / 84
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 informationThere are five problems. Solve four of the five problems. Each problem is worth 25 points. A sheet of convenient formulae is provided.
Preliminary Examination (Solutions): Partial Differential Equations, 1 AM - 1 PM, Jan. 18, 16, oom Discovery Learning Center (DLC) Bechtel Collaboratory. Student ID: There are five problems. Solve four
More informationAn Introduction to Algebraic Multigrid (AMG) Algorithms Derrick Cerwinsky and Craig C. Douglas 1/84
An Introduction to Algebraic Multigrid (AMG) Algorithms Derrick Cerwinsky and Craig C. Douglas 1/84 Introduction Almost all numerical methods for solving PDEs will at some point be reduced to solving A
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 informationTD 1: Hilbert Spaces and Applications
Université Paris-Dauphine Functional Analysis and PDEs Master MMD-MA 2017/2018 Generalities TD 1: Hilbert Spaces and Applications Exercise 1 (Generalized Parallelogram law). Let (H,, ) be a Hilbert space.
More informationChapter 1 Piecewise Polynomial Approximation in 1D
Chapter 1 Piecewise Polynomial Approximation in 1D Abstract n this chapter we introduce a type of functions called piecewise polynomials that can be used to approximate other more general functions, and
More informationNumerical Methods for Partial Differential Equations
Numerical Methods for Partial Differential Equations Eric de Sturler University of Illinois at Urbana-Champaign Read section 8. to see where equations of type (au x ) x = f show up and their (exact) solution
More informationContents. Prologue Introduction. Classical Approximation... 19
Contents Prologue........................................................................ 15 1 Introduction. Classical Approximation.................................. 19 1.1 Introduction................................................................
More informationMAE 107 Homework 7 Solutions
MAE 107 Homework 7 Solutions 1. Tridiagonal system for u xx (x) + u(x) = 1 + sin(πx/4), for x [0, ], where u(0) =, u() = 4, with step size h = 1/. The total number of segments are: n = 0 1/ =. The nodes
More informationACM/CMS 107 Linear Analysis & Applications Fall 2017 Assignment 2: PDEs and Finite Element Methods Due: 7th November 2017
ACM/CMS 17 Linear Analysis & Applications Fall 217 Assignment 2: PDEs and Finite Element Methods Due: 7th November 217 For this assignment the following MATLAB code will be required: Introduction http://wwwmdunloporg/cms17/assignment2zip
More informationNonlinear, Transient Conduction Heat Transfer Using A Discontinuous Galerkin Hierarchical Finite Element Method
Nonlinear, Transient Conduction Heat Transfer Using A Discontinuous Galerkin Hierarchical Finite Element Method by Jerome Charles Sanders B.S. in Physics, May 2002 The College of New Jersey A Thesis submitted
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 informationBoundary Value Problems and Iterative Methods for Linear Systems
Boundary Value Problems and Iterative Methods for Linear Systems 1. Equilibrium Problems 1.1. Abstract setting We want to find a displacement u V. Here V is a complete vector space with a norm v V. In
More informationSome Notes on Elliptic Regularity
Some Notes on Elliptic Regularity Here we consider first the existence of weak solutions to elliptic problems of the form: { Lu = f, u = 0, (1) and then we consider the regularity of such solutions. The
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 informationOptimal multilevel preconditioning of strongly anisotropic problems.part II: non-conforming FEM. p. 1/36
Optimal multilevel preconditioning of strongly anisotropic problems. Part II: non-conforming FEM. Svetozar Margenov margenov@parallel.bas.bg Institute for Parallel Processing, Bulgarian Academy of Sciences,
More informationLecture 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 informationSchwarz Preconditioner for the Stochastic Finite Element Method
Schwarz Preconditioner for the Stochastic Finite Element Method Waad Subber 1 and Sébastien Loisel 2 Preprint submitted to DD22 conference 1 Introduction The intrusive polynomial chaos approach for uncertainty
More informationSpline Element Method for Partial Differential Equations
for Partial Differential Equations Department of Mathematical Sciences Northern Illinois University 2009 Multivariate Splines Summer School, Summer 2009 Outline 1 Why multivariate splines for PDEs? Motivation
More informationThe Spectral-Element Method: Introduction
The Spectral-Element Method: Introduction Heiner Igel Department of Earth and Environmental Sciences Ludwig-Maximilians-University Munich Computational Seismology 1 / 59 Outline 1 Introduction 2 Lagrange
More informationAlgorithms for Scientific Computing
Algorithms for Scientific Computing Finite Element Methods Michael Bader Technical University of Munich Summer 2016 Part I Looking Back: Discrete Models for Heat Transfer and the Poisson Equation Modelling
More informationLecture 10 Polynomial interpolation
Lecture 10 Polynomial interpolation Weinan E 1,2 and Tiejun Li 2 1 Department of Mathematics, Princeton University, weinan@princeton.edu 2 School of Mathematical Sciences, Peking University, tieli@pku.edu.cn
More information2 Elliptic Differential Equations
2 Elliptic Differential Equations 2.1 Classical solutions As far as existence and uniqueness results for classical solutions are concerned, we restrict ourselves to linear elliptic second order elliptic
More informationWavelets in Scattering Calculations
Wavelets in Scattering Calculations W. P., Brian M. Kessler, Gerald L. Payne polyzou@uiowa.edu The University of Iowa Wavelets in Scattering Calculations p.1/43 What are Wavelets? Orthonormal basis functions.
More informationSolving the steady state diffusion equation with uncertainty Final Presentation
Solving the steady state diffusion equation with uncertainty Final Presentation Virginia Forstall vhfors@gmail.com Advisor: Howard Elman elman@cs.umd.edu Department of Computer Science May 6, 2012 Problem
More informationStrauss PDEs 2e: Section Exercise 2 Page 1 of 6. Solve the completely inhomogeneous diffusion problem on the half-line
Strauss PDEs 2e: Section 3.3 - Exercise 2 Page of 6 Exercise 2 Solve the completely inhomogeneous diffusion problem on the half-line v t kv xx = f(x, t) for < x
More informationMathematical methods for engineering. The finite element method
. The finite element method University of Trento Trento Winter School on Numerical Methods February 20 th - March 3 rd, 2017 FEM: Remarks on programming Model problem: with a o 0. u + a 0 u = f u = 0 u
More informationParallel Discontinuous Galerkin Method
Parallel Discontinuous Galerkin Method Yin Ki, NG The Chinese University of Hong Kong Aug 5, 2015 Mentors: Dr. Ohannes Karakashian, Dr. Kwai Wong Overview Project Goal Implement parallelization on Discontinuous
More informationXingye Yue. Soochow University, Suzhou, China.
Relations between the multiscale methods p. 1/30 Relationships beteween multiscale methods for elliptic homogenization problems Xingye Yue Soochow University, Suzhou, China xyyue@suda.edu.cn Relations
More informationSmall zeros of hermitian forms over quaternion algebras. Lenny Fukshansky Claremont McKenna College & IHES (joint work with W. K.
Small zeros of hermitian forms over quaternion algebras Lenny Fukshansky Claremont McKenna College & IHES (joint work with W. K. Chan) Institut de Mathématiques de Jussieu October 21, 2010 1 Cassels Theorem
More information44 CHAPTER 3. CAVITY SCATTERING
44 CHAPTER 3. CAVITY SCATTERING For the TE polarization case, the incident wave has the electric field parallel to the x 3 -axis, which is also the infinite axis of the aperture. Since both the incident
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 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 informationSINC PACK, and Separation of Variables
SINC PACK, and Separation of Variables Frank Stenger Abstract This talk consists of a proof of part of Stenger s SINC-PACK computer package (an approx. 400-page tutorial + about 250 Matlab programs) that
More informationSolution to Set 7, Math 2568
Solution to Set 7, Math 568 S 5.: No. 18: Let Q be the set of all nonsingular matrices with the usual definition of addition and scalar multiplication. Show that Q is not a vector space. In particular,
More informationFinite Element Analysis Prof. Dr. B. N. Rao Department of Civil Engineering Indian Institute of Technology, Madras. Module - 01 Lecture - 11
Finite Element Analysis Prof. Dr. B. N. Rao Department of Civil Engineering Indian Institute of Technology, Madras Module - 01 Lecture - 11 Last class, what we did is, we looked at a method called superposition
More informationContinuous and Discontinuous Finite Element Methods for a Peridynamics Model of Mechanics DRAFT
Continuous and Discontinuous Finite Element Methods for a Peridynamics Model of Mechanics Xi Chen and Max Gunzburger Department of Scientific Computing, Florida State University, Tallahassee FL 3306-40,
More informationVariational Formulation of Plane Beam Element
13 Variational Formulation of Plane Beam Element IFEM Ch 13 Slide 1 Beams Resist Primarily Transverse Loads IFEM Ch 13 Slide 2 Transverse Loads are Transported to Supports by Flexural Action Neutral surface
More informationarxiv: v1 [math.na] 1 May 2013
arxiv:3050089v [mathna] May 03 Approximation Properties of a Gradient Recovery Operator Using a Biorthogonal System Bishnu P Lamichhane and Adam McNeilly May, 03 Abstract A gradient recovery operator based
More informationComputational Methods for Engineering Applications II. Homework Problem Sheet 4
S. Mishra K. O. Lye L. Scarabosio Autumn Term 2015 Computational Methods for Engineering Applications II ETH Zürich D-MATH Homework Problem Sheet 4 This assignment sheet contains some tasks marked as Core
More informationIntroduction to PDEs: Assignment 9: Finite elements in 1D
Institute of Scientific Computing Technical University Braunschweig Dr. Elmar Zander Dr. Noemi Friedman Winter Semester 213/14 Assignment 9 Due date: 6.2.214 Introduction to PDEs: Assignment 9: Finite
More informationMATH 333: Partial Differential Equations
MATH 333: Partial Differential Equations Problem Set 9, Final version Due Date: Tues., Nov. 29, 2011 Relevant sources: Farlow s book: Lessons 9, 37 39 MacCluer s book: Chapter 3 44 Show that the Poisson
More informationFinite Difference and Finite Element Methods
Finite Difference and Finite Element Methods Georgy Gimel farb COMPSCI 369 Computational Science 1 / 39 1 Finite Differences Difference Equations 3 Finite Difference Methods: Euler FDMs 4 Finite Element
More informationChapter 5 A priori error estimates for nonconforming finite element approximations 5.1 Strang s first lemma
Chapter 5 A priori error estimates for nonconforming finite element approximations 51 Strang s first lemma We consider the variational equation (51 a(u, v = l(v, v V H 1 (Ω, and assume that the conditions
More informationRitz Method. Introductory Course on Multiphysics Modelling
Ritz Method Introductory Course on Multiphysics Modelling TOMASZ G. ZIELIŃSKI bluebox.ippt.pan.pl/ tzielins/ Institute of Fundamental Technological Research of the Polish Academy of Sciences Warsaw Poland
More informationRecovery Techniques in Finite Element Methods
Chapter 8 Recovery Techniques in Finite Element Methods Zhimin Zhang Department of Mathematics Wayne State University Detroit, MI 48202 USA zhang@math.wayne.edu Contents 8. Introduction and preliminary.............................
More informationOutline. 1 Boundary Value Problems. 2 Numerical Methods for BVPs. Boundary Value Problems Numerical Methods for BVPs
Boundary Value Problems Numerical Methods for BVPs Outline Boundary Value Problems 2 Numerical Methods for BVPs Michael T. Heath Scientific Computing 2 / 45 Boundary Value Problems Numerical Methods for
More informationAn additive average Schwarz method for the plate bending problem
J. Numer. Math., Vol. 10, No. 2, pp. 109 125 (2002) c VSP 2002 Prepared using jnm.sty [Version: 02.02.2002 v1.2] An additive average Schwarz method for the plate bending problem X. Feng and T. Rahman Abstract
More informationNumerical Solutions to Partial Differential Equations
Numerical Solutions to Partial Differential Equations Zhiping Li LMAM and School of Mathematical Sciences Peking University Introduction to Hyperbolic Equations The Hyperbolic Equations n-d 1st Order Linear
More informationIntroduction to Spectral Element Methods
Introduction to Sectral Element Methods by Haijun Yu Email: yuhaijun@gmail.com One-Dimensional Sectral Element Method We use the -dimensional ellitic Helmholtz equation as a model equation. αu u f, x I0,L,
More informationSystem of Linear Equations
Math 20F Linear Algebra Lecture 2 1 System of Linear Equations Slide 1 Definition 1 Fix a set of numbers a ij, b i, where i = 1,, m and j = 1,, n A system of m linear equations in n variables x j, is given
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 informationQualifying Examination
Summer 24 Day. Monday, September 5, 24 You have three hours to complete this exam. Work all problems. Start each problem on a All problems are 2 points. Please email any electronic files in support of
More information