By Krebs A., Stephan E.P.

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**Additional resources for A p-version finite element method for nonlinear elliptic variational inequalities in 2D**

**Example text**

Both M and A have the same small eigenvalues). e. |a| for a ∈ R. The same can be done for vectors in Rn and matrices in Rm×n , through a non-negative measure called the norm. The following definition, using three norm axioms, determines if any such nonnegative measure is a norm. 2. Let V be either Rn (for vectors) or Rm×n (for matrices). A measure u of u ∈ V , satisfying the following Norm axioms, is a valid norm: r u ≥ 0 for any u and u = 0 is and only if u = 0, r αu = |α| u for any u and any α ∈ R, r u + v ≤ u + v for any u, v ∈ V .

Here a b = (a − 1) Mod b + 1 = mod(a − 1, b) + 1. 13). For example, 256 256 = 256 and 257 256 = 1. 1), transforming column and row vectors, respectively, will be A = W AW T . 49) without forming W explicitly. Thus the DWT can be done in a fast way so it can ‘automatically’ become the FWT. 47) can be estimated by Tv = mn + mn/2 + · · · + mn/2t−1 ≈ 2mn Step 1 Step 2 Step t and for a dense matrix Tm = 2mn 2 + 2mnn/2 + · · · + 2mnn/2t−1 ≈ 4mn 2 . 50) ˘ νT W Jν n×n and c0 cm−2 · · · . cm−1 . .. c2 c1 c3 ..

E. Pn = I (:, rn + 1) or PnT = I (rn + 1, :). For a small n, one can afford to implement each permutation step by step to find rn . 6 Fast Fourier transforms and fast wavelet transforms 25 the better method for getting rn is the so-called bit reversal algorithm [465]. Let (b0 b1 . . bt−1 )2 be the binary representation of integer k ∈ [0, n − 1]. Here exactly t binary digits are needed, due to the fact (111 . . 11)2 = 20 + 21 + · · · + 2t−1 = 2t − 1 = n − 1. Then by induction [465], one can show that the kth index in the permutation vector rn satisfies rn (k) = (bt−1 bt−2 · · · b1 b0 )2 .

### A p-version finite element method for nonlinear elliptic variational inequalities in 2D by Krebs A., Stephan E.P.

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