64b is established with c1 2 2 the remainder of the

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Unformatted text preview: being the number of nodes on the standard triangle. However, with minor alterations, the results apply to other bases and, indeed, other element shapes. We proceed with one preliminary theorem and then present the main result. Theorem 4.6.2. Let p be the largest integer for which the interpolant (4.6.7) is exact when u( ~ ) is a polynomial of degree p. Then, there exists a constant C > 0 such that ju ; U js ~~ 0 C jujp+1 ~ 0 8u 2 H p+1( 0 ) s = 0 1 : : : p + 1: (4.6.8) Proof. The proof utilizes the Bramble-Hilbert Lemma and is presented in Axelsson and Barker 2]. Theorem 4.6.3. Let be a polygonal domain that has been discretized into a net of triangular elements e , e = 1 2 : : : N . Let h and denote the largest element edge and smallest angle in the mesh, respectively. Let p be the largest integer for which (4.6.7) is exact when u( ) is a complete polynomial of degree p. Then, there exists a constant ~ C > 0, independent of u 2 H p+1 and the mesh, such that ju ; U js C hp+1;s juj sin ]s p+1 8u 2 H p+1 ( ) s = 0 1: (4.6.9) Remark 1. The results are restricted s = 0 1 because, typically, U 2 H 1 \ H p+1 . 34 Finite Element Approximation Proof. Consider an element e and use the left inequality of (4.6.4a) with replaced by u ; U to obtain ju ; U j2 e c;2 sin;2s+1 e h;2s+2 ju ; U j2 0 : ~ ~s s s e Next, use (4.6.8) ju ; U j2 e c;2 sin;2s+1 e h;2s+2 C juj2+1 0 : ~p s s e Finally, use the right inequality of (4.6.4a) to obtain ju ; U j2 s 2 c;2 sin;2s+1 eh;2s+2CCp+1 sin;1 eh2pjuj2+1 e : s e ep e Combining the constants ju ; U j2 s C sin;2s eh2(p+1;s)juj2+1 e : e p Summing over the elements and taking a square root gives (4.6.9). A similar result for rectangles follows. e Theorem 4.6.4. Let the rectangular domain be discretized into a mesh of rectangular elements e, e = 1 2 : : : N . Let h and denote the largest element edge and smallest edge ratio in the mesh, respectively. Let p be the largest integer for which (4.6.7) is exact when u( ) is a complete polynomial of degree p. Then, there exists a constant C > 0, ~ independent of u 2 H p+1 and the mesh, such that ju ; U js C hp+1;s juj s p+1 8u 2 H p+1( ) s = 0 1: (4.6.10) Proof. The proof follows the lines of Theorem 4.6.3 2]. Thus, small and large (near ) angles in triangular meshes and small aspect ratios (the minimum to maximum edge ratio of an element) in a rectangular mesh must be avoided. If these quantities remain bounded then the mesh is uniform as expressed by the following de nition. De nition 4.6.1. A family of nite element meshes h is uniform if all angles of all elements are bounded away from 0 and and all aspect ratios are bounded away from zero as the element size h ! 0. With such uniform meshes, we can combine Theorems 4.6.2, 4.6.3, and 4.6.4 to obtain a result that appears more widely in the literature. Theorem 4.6.5. Let a family of meshes h be uniform and let the polynomial interpolant U of u 2 H p+1 be exact whenever u is a complete polynomial of degree p. Then there exists a constant C > 0 such that ju ; U j s C hp+1;sjujp+1 s = 0 1:...
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