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1 Model problem 1.1 Poisson Equation in 1D Boundary Value Problem(BVP) (x)=∫(x) (0,1),u(0)=(1)=0,f Describes many simple physical phenomena(e.g) Deformation of an elastic bar Deformation of a string under tension Temperature distribution in a bar The Poisson equation in one dimension is in fact an ordinary differ tion. When dealing with ordinary differential equations we Poisson equation will be used here to illastrate numerical techniques for elliptic PDE's in multi-dimensions. Other techniques specialized for ordinary differen tial equations could be used if we were only interested in the one dimension

Course Outline Overview of pdes(1) o Finite differences methods(6) Finite volume methods(3) Finite element methods(7) Boundary integral methods(6)

Despite its apparent simplicity this equation appears in a wide range of dis m heat 7 to financial er we will make extensive use of this equation, and several of the limiting cases contained therein, to illustrate the numerical techniques that will be presented

Background Developed over the last 25 years- Brandt (1973) published first paper with practical results Offers the possibility of solving a problem with work and storage proportional to the number of unknowns Well developed for linear elliptic problems application to other equations is still an active area of research

1 First Order ave Equation SLIDE 1 The simplest first order partial differential equation in two variables(a, t)is the linear wave equation. Recall that all first order PDE's are of hyperbolic type INITIAL BOUNDARY VALUE PROBLEM (IBVP) 0,x∈(0,1)

Motivation Consider a standard second order finite difference discretization

1 Background Brandt(1973)published first paper SLIDE 1 Offers the possibility of solving a problem with work and storage propor tional to the number of unknowns Well developed for elliptic proble

1 Motivation SLIDE 1 Consider a standard second order finite difference discretization of V-u= on a regular g 1.2. and 3 dimensions 1.1 1D Finite differences

Governing Equation Stability Analysis 3 Examples Relationship between σ and λh Implicit Time-Marching Scheme