The completion of locally refined simplicial partitions created by bisection 论文
摘要
Recently, in [Found. Comput. Math., 7(2) (2007), 245–269], we proved that an adaptive finite element method based on newest vertex bisection in two space dimensions for solving elliptic equations, which is essentially the method from [ <italic>SINUM</italic> , 38 (2000), 466–488] by Morin, Nochetto, and Siebert, converges with the optimal rate.The number of triangles <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper N"> <mml:semantics> <mml:mi>N</mml:mi> <mml:annotation encoding="application/x-tex">N</mml:annotation> </mml:semantics> </mml:math> </inline-formula> in the output partition of such a method is generally larger than the number <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper M"> <mml:semantics> <mml:mi>M</mml:mi> <mml:annotation encoding="application/x-tex">M</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of triangles that in all intermediate partitions have been marked for bisection, because additional bisections are needed to retain conforming meshes.A key ingredient to our proof was a result from [ <italic>Numer. Math.</italic> , 97(2004), 219–268] by Binev, Dahmen and DeVore saying that <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper N minus upper N 0 less-than-or-equal-to upper C upper M"> <mml:semantics> <mml:mrow> <mml:mi>N</mml:mi> <mml:mo> − </mml:mo> <mml:msub> <mml:mi>N</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:mo> ≤ </mml:mo> <mml:mi>C</mml:mi> <mml:mi>M</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">N-N_0 \leq C M</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for some absolute constant <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper C"> <mml:semantics> <mml:mi>C</mml:mi> <mml:annotation encoding="application/x-tex">C</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , where <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper N 0"> <mml:semantics> <mml:msub> <mml:mi>N</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:annotation encoding="application/x-tex">N_0</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is the number of triangles from the initial partition that have never been bisected. In this paper, we extend this result to bisection algorithms of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n"> <mml:semantics> <mml:mi>n</mml:mi> <mml:annotation encoding="application/x-tex">n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -simplices, with that generalizing the result concerning optimality of the adaptive finite element method to general space dimensions.