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Course: M 610, Fall 2009
School: Texas A&M
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Word Count: 415

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#3. Homework (Due Feb. 16) Question 1 [Bramble-Hilbert Lemma] Let E be a subset of Rd such that the hypotheses of the Deny-Lions Lemma hold (E is non empty, connected, open, and Lipschitz). Let k be a nonnegative integer. Let f be a bounded linear mapping from H k+1 (E) to R. Assume that Pk (i.e. set of the polynomials in d variable, of total degree at most k) is in the null space of f . Prove that there exists c...

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#3. Homework (Due Feb. 16) Question 1 [Bramble-Hilbert Lemma] Let E be a subset of Rd such that the hypotheses of the Deny-Lions Lemma hold (E is non empty, connected, open, and Lipschitz). Let k be a nonnegative integer. Let f be a bounded linear mapping from H k+1 (E) to R. Assume that Pk (i.e. set of the polynomials in d variable, of total degree at most k) is in the null space of f . Prove that there exists c > 0 such that v H k+1 (E), Question 2 Let E be a subset of Rd (d 3) such that the hypotheses of the DenyLions Lemma hold (E is non empty, connected, open, and Lipschitz). Let {1 , . . . , n } E and {1 , . . . , n } R be a set of nodes and a set of weights so that the following quadrature rule holds true n |f (v)| c |v|H k+1 (E) . p Pk , E p(x)dx = i=1 i p(i ), where k is a positive integer. Admit the fact that there exists c0 > 0 so that for all functions in C 0 (E), maxxE |(x)| c0 H 2 (E) . Prove that there exists c > 0 such that n v H k+1 (E), | E v(x)dx i=1 v(i E i )| c|v|H k+1 (E) . v(x)dx n i=1 i v(i ).) (Hint: Consider the mapping f : v Question 3 Let K be an element in a mesh of a domain in Rd , d 3. Let K be the reference element. Assume that there is an ane and isomorphic mapping T : K K. Denote by J the Jacobian matrix of T . Let {1 , . . . , n } K and {1 , . . . , n } R be a set of nodes and a set of weights so that the following quadrature rule holds true n p Pk , K p(x)dx = i=1 1 i p(i ), 2 where k is a positive integer. Set i = T (i ), and i = det(J)i , 1 i n. Prove that there exists c > 0 (depending only on K and k)...

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