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We will construct a discrete model whose solution gives an accurate ap-proximation to the exact solution at a finite collection of selected points(called nodes) in [0,1]. We take the nodes to be equally spaced and to in-clude the interval end points (boundary). So we choose a positive integerNand define the nodes or grid pointsx0= 0, x1=h, x2= 2h, . . . , xN=Nh, xN+1= 1,(10.64)whereh= 1/(N+ 1) is thegrid sizeor node spacing. The nodesx1, . . . , xNare called interior nodes, because they lie inside the interval [0,1], and thenodesx0andxN+1are called boundary nodes.We now construct a discrete approximation to the ordinary differentialequation by replacing the second derivative with a second order finite differ-ence approximation. As we know,u00(xj) =u(xj+1-2u(xj) +u(xj-1h2+O(h2).(10.65)Neglecting theO(h2) error and denoting the approximation ofu(xj) byvj(i.e.vj≈u(xj))fj=f(xj) andcj=c(xj), forj= 1, . . . , N, then at eachinterior node-vj-1+ 2vj+vj+1h2+cjvj=fj,j= 1,2, . . . , N(10.66)
170CHAPTER 10.LINEAR SYSTEMS OF EQUATIONS Iand at the boundary nodes, applying (10.63), we havev0=vN+1= 0.(10.67)Thus, (10.66) is a linear system ofNequations inNunknownsv1, . . . , vN,which we can write in matrix form as1h22 +c1h2-10· · ·· · ·0-12 +c2h2-1......0..........................................0.........-10· · ·0-12 +cNh2v1v2............vN=f1f2............fN.(10.68)The matrix, let us call itA, of this system is tridiagonal and symmetric.A direct computation shows that for an arbitrary, nonzero, column vectorv= [v1, . . . , vN]TvTAv=NXj=1"vj+cjvjh2+cjv2j#>0,∀v6= 0(10.69)and therefore, sincecj≥0 for allj,Ais positive definite. Thus, there is aunique solution to (10.68) and can be efficiently found with our tridiagonalsolver, Algorithm 5. Since the expected numerical error isO(h2) =O(1/(N+1)2), even a modest accuracy ofO(10-4) requiresN≈100.10.6A 2D BVP: Dirichlet Problem for thePoisson’s EquationWe now look at a simple 2D BVP for an equation that is central to manyapplications, namely Poisson’s equation. For concreteness here, we can thinkof the equation as a model for small deformationsuof a stretched, squaremembrane fixed to a wire at its boundary and subject to a force density
10.6. A 2D BVP: DIRICHLET PROBLEM FOR THE POISSON’S EQUATION171f. Denoting by Ω, and∂Ω, the unit square [0,1]×[0,1] and its boundary,respectively, the BVP is to findusuch that-Δu(x, y) =f(x, y),for (x, y)∈Ω(10.70)andu(x, y) = 0.for (x, y)∈∂Ω(10.71)In (10.70), Δuis the Laplacian of u, also denoted as∇2u, and is given byΔu=∇2u=uxx+uyy=∂2u∂x2+∂2u∂y2.(10.72)Equation (10.70) is Poisson’s equation (in 2D) and together with (10.71)specify a (homogeneous) Dirichlet problem because the value ofuis given atthe boundary.