ProgrEx2 - Department of Mathematics University of Houston...

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Unformatted text preview: Department of Mathematics University of Houston Numerical Analysis I Dr. Ronald H.W. Hoppe Numerical Analysis I (2nd Practical Homework Assignment) Practical Exdercise 2 ( cg method ) The deflection u = u ( x ) , x = ( x 1 ,x 2 ) T ∈ Ω := (0 , 1) 2 of a clamped membrane due to an exterior force of force density f = f ( x ) , x ∈ Ω can be described by the boundary value problem- Δ u ( x ) =- ( ∂ ∂x 2 1 u ( x ) + ∂ ∂x 2 2 u ( x )) = f ( x ) , x ∈ Ω u ( x ) = 0 , x ∈ Γ := ∂ Ω . The approximation of the 2D Laplacian- Δ := ∂ ∂x 2 1 + ∂ ∂x 2 2 by finite differences ∂ ∂x 2 1 u ( x ) ∼ u ( x 1- h,x 2 )- 2 u ( x 1 ,x 2 ) + u ( x 1 + h,x 2 ) h 2 , ∂ ∂x 2 2 u ( x ) ∼ u ( x 1 ,x 2- h )- 2 u ( x 1 ,x 2 ) + u ( x 1 ,x 2 + h ) h 2 with respect to a uniform grid of mesh size h > 0 results in a linear algebraic system with the coefficient matrix A = ( a ij ) n- 1 i,j =0 ∈ lR n × n ,n = k 2 , k ∈ lN, which is given by a ij =...
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ProgrEx2 - Department of Mathematics University of Houston...

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