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
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
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)
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1 Motivation The Poisson problem has a strong formulation a minimization formulation and a weak formulation T weak formulations are more general than the strong formulation in terms of regularity and admissible data SLIDE 2 The minimization/weak formulations are defined by: a space X; a bilinear The minimization/weak formulations identify ESSENTIAL boundary conditions NATURAL boundary conditions ed in a The points of departure for the finite element method are the weak formulation(more generally) the minimization statement (if a is SPD) 2 The dirichlet problem 2.1 Strong Formulation Find u such that
