Mathematical Physics - Volume II - Numerical Methods

3.5 Finite element approximations

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1. Stiffness matrix K is sparse, since global functions φ i and φ j and their derivatives are not equal to zero only when there are elements that contain both nodes i and j . 2. In the case being analysed, K is a symmetrical matrix (since the operator in (3.46) is self adjoined) and has the form of a strip in the case that the nodes were numbered in order. All of this allows for efficient solving of linear algebraic equation system (3.88). 3. All integrals in (3.89) and (3.90) can be calculated as the sum of all mesh elements contribu tions, for which the exact solution of a boundary problem satisfies the condition given by the following equation: Z Ω e k ∇ u · ∇ v d x d y − Z Ω e f v d x d y + Z ∂ Ω e ˆ σ n v d s = 0 (3.91) for each allowed v , where ˆ σ n is the normal flux component at the element boundary. Let u e h and v e h represent the boundaries of approximations u h and v h on Ω e . Then, the local approximation of the variation boundary condition on Ω e has the following form: Z Ω e k ∇ u h · ∇ v h d x d y − Z Ω e f v h d x d y + Z ∂ Ω e ˆ σ n v h d s = 0 , (3.92) where ˆ σ n represents the real (exact) flux through ∂ Ω e , which, despite not being given as data in the initial problem, still occurs as a natural boundary condition at Ω e . Since v h = 0 at ∂ Ω 1 h , there will be no contribution to the last integral in (3.91) by the elements whose sides coincide with ∂ Ω 1 h . Since u e h and v e h have the following form: u e h ( ξ , η ) = u e i ψ e i ( ξ , η ) v e h ( ξ , η ) = v e i ψ e i ( ξ , η ) , i = 1 , . . . , N e (3.93) where ψ e i are local shape functions for Ω e , and N e is the number of nodes in Ω e . Equation (3.91) now becomes a linear system: k e i j u e j = f e i − σ e i , i , j = 1 , 2 , . . . , N e (3.94) where k e i j = Z Ω e k ∇ ψ e i · ∇ ψ e j d x d y . (3.95) f e i = Z Ω e f ψ e i d x d y . (3.96) Stiffness matrix and load vector components of element Ω e whereas the following expression: σ e i = Z Ω e σ n ψ e i d s (3.97) Formally, global system of equations (3.88) is obtained by summing of (3.94) with respect to all elements in the mesh, wherein matrix and vector terms (3.95), (3.96) and (3.97) are placed in their corresponding locations in the global system matrix. As an example, stiffness matrix terms for element Ω e are placed in columns and rows which correspond to that element’s nodes. Thus, first terms in expressions (3.89) and (3.90) are obtained as: E ∑ e = 1 Z Ω e k ∇ φ i · ∇ φ j d x d y = E ∑ e = 1 K e i j (3.98) E ∑ e = 1 Z Ω e f φ i d x d y = E ∑ e = 1 F e i i , j = 1 , 2 , · · · , N , (3.99)