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These results and those of strouboulis and haque 19

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Unformatted text preview: een error estimates for odd- and even-order nite element solutions involves a profound, but little known, result of Babuska (cf. 1, 2, 3, 9, 22, 23]). Concentrating on linear second-order elliptic problems on rectangular meshes, Babuska indicates that asymptotically (as mesh spacing tends to zero) errors of odd-degree nite element solutions occur near element edges while errors of even-degree solutions occur in element interiors. These ndings suggest that error estimates may be obtained by neglecting errors in element interiors for odd-degree polynomials and neglecting errors on element boundaries for even-degree polynomials. Thus, for piecewise odd-degree approximations, we could neglect the area integrals on the right-hand sides of (7.4.8) or (7.4.9a) and calculate an error estimate by solving + ; Ae(V E ) =< V (pUn ) + (pUn ) >e 2 ~ 8V 2 S N : (7.4.16a) + ; Ae(V E ) =< V (pUn) ; (pUn) >e 2 ~ 8V 2 S N : (7.4.16b) or For piecewise even-degree approximations, the boundary terms in (7.4.8) or (7.4.9a) can be neglected t...
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