Mathematical Physics - Volume II - Numerical Methods

Chapter 3. Comparison of finite element method and finite difference method

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adjacent elements, the continuity of the function on their common side is achieved with the condition that the linear functions of these elements have the same values in common nodes. Similar to this, the linear function given below: v h = a 1 + a 2 ξ + a 2 η + a 3 ξη (3.65) has four constants, and thus its corresponding finite element is rectangular, and its quadratic function is: v h = a 1 + a 2 ξ + a 3 η + a 4 ξ 2 + a 5 ξη + a 6 η 2 (3.66) which has 6 constants, and requires a six node triangle (with three nodes at its points, and three more at each side midpoint). In accordance with the description given in the previous chapter, we can now define interpolation g h for function g in the following form: g h ( ξ , η ) = g i φ i ( ξ , η ) ( ξ , η ) ∈ Ω h (3.67) where φ i , i = 1 , · · · , N are the base functions defined at Ω h as: φ i ( ξ j , η j ) = 1 , za i = j 0 , za i̸ = j (3.68) (3.69) According to the above, by including g j = g ( ξ j , η j ) , g j becomes equal to the value of g in nodes, i.e. it becomes its interpolation. The following two conditions also need to be fulfilled: 1. Definition of local interpolation functions ψ e i for each element must be such that its fitting in the finite element mesh results in base functions which fulfill (3.68); 2. Base functions φ i should be square-integrable, along with their partial derivatives: Z Ω h " ∂φ i ∂ x 2 + ∂φ i ∂ y 2 + φ 2 i # d x d y < ∞ . (3.70) This requirement is fulfilled if functions φ i are continual along the element boundaries. (3.71) defines a planar surface, linear interpolation within a triangle approximates a smooth function (surface) v with plane (3.71), inside an arbitrary element Ω e . Let us assume that Ω h is made of a total of E triangular elements, and that the following linear interpolation holds: v e h ( ξ , η ) = a 1 + a 2 ξ + a 3 η . (3.72) The following applies to triangle nodes (fig. 3.10): v 1 = v e h ( ξ 1 , η 1 ) = a 1 v 2 = v e h ( ξ 2 , η 2 ) = a 1 + a 2 v 3 = v e h ( ξ 3 , η 3 ) = a 1 + a 3 (3.73) where ( ξ j , η j ) are finite element nodes, which means that the following holds: g h ( ξ j , η j ) = g j j = 1 , 2 , · · · , N . v h ( ξ , η ) = a 1 + a 2 ξ + a 3 η

3.4.1 Interpolation within triangles Since the linear function: