,. .. and .. .. , n k we can find a linear combination ??F K ?K,?,q i that has the same moments of degree ? k ? 1 as ?K,i on each ? ? FK . The function ?K,i ? ??F K ?K,?,q i therefore has zero moments of degree ? k ? 1 on each face and, extended by 0 outside K, satisfies the (k ? 1)-degree patch test: its moments on each face coincide when viewed from each side of the faces. When {K, L} = M?, for a given q ? P k?1 (?), by (56) the functions ?K,?,q and ?L,?,q have the same moments of degree ? k ? 1 on ?. Hence, in a similar way as in (23), we can glue ?K,?,q and ?L,?,q to obtain a global function that satisfies the (k ? 1)-degree patch test. The family of these extended functions span a non-conforming space that has approximation properties of order k

.. J=1, k be a basis of P k?1 (?), the family {?K,i : i = 1, . . . , nK } ? {?K,?,q j : ? ? FK, The only caveat is the following: letting (qj)

, Hence, describing a space of the local space (and, in consequence, the global space) requires to actually solve local linear problems, extracting a basis from a generating family, general, this family is linearly independent

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