Mathematical Physics - Volume II - Numerical Methods

3.5 Finite element approximations

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It is clear that φ i fulfill the condition given by (3.77) and that they are continuous along the element boundaries, i.e. at Ω h ; their first partial derivatives are reduced to step functions, hence their squares are also integrable. Thus, such base functions represent an adequate choice for finite element approximation applications. 3.4.2 Interpolation error Let us assume that a smooth function g is interpolated by function g h , which contains complete polynomials of the k -th order. If partial derivatives of the k + 1-th order of function g are limited within Ω e , the following applies to the interpolation error: ∥ g − g h ∥ ∞ , Ω e = max | g ( x , y ) − g h ( x , y ) | < Ch k + 1 e , (3.80) where C is a positive constant, and h e is the “diameter” of Ω e , i.e. the greatest distance between any two points in the element. Similar to this, we have that: ∂ g ∂ x − ∂ g h ∂ x ∞ , Ω e ≤ C 1 h k e i ∂ g ∂ y − ∂ g h ∂ y ∞ , Ω e ≤ C 2 h k e . (3.81) If we assume that Ω h = Ω and that h is the greatest “diameter” of all elements, we obtain the following (assuming that the mesh is sufficiently regular): ∥ g − g h ∥ 1 ≤ C 3 h k (3.83) for a sufficiently small h . In this case, it is also necessary for g h to contain a complex polynomial of the k -th order. 3.5 Finite element approximations Let H 1 ( Ω ) be a class of functions which satisfy expression H 1 ( Ω ) , and are defined for the whole domain Ω . The problem here involves the determining of a function u from H 1 ( Ω ) , such that u = ˆ u at boundary ∂ Ω 1 and that the following condition is fulfilled: Z Ω k ∇ u · ∇ v d x d y + Z ∂ Ω 2 ˆ σ v d s − Z Ω f v d x d y = 0 (3.84) for ∀ v ∈ H 1 ( Ω ) , such that v = 0 at ∂ Ω 1 and γ = p ˆ u . The approximate solution of equation (3.84) requires replacing of domain Ω with Ω h , which is actually the finite element mesh (with a total of N nodes and E elements), and defining of an N -dimensional subspace H h in H 1 ( Ω h ) by introducing global base functions φ i , i = 1 , · · · , N , along with finite elements. In order fr these shape functions to remain continuous along each element, they cannot be used to model jumps, e.g. a jump in material properties. Thus the finite element mesh is generated in a way that ensures that element nodes and edges correspond to the lines where the material modulus jump occurs, Figure 3.13. H 1 is the two-dimensional norm defined by the following expression: ∥ g ∥ 2 1 = Z Ω " g 2 + ∂ g ∂ x 2 + ∂ g ∂ y 2 # d x d y . (3.82)