On the convergence rate of the cell discretization algorithm for solving elliptic problems

Abstract

Error estimates for the cell discretization algorithm are obtained for polynomial bases used to approximate both <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper H Superscript k Baseline left-parenthesis normal upper Omega right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>H</mml:mi> <mml:mi>k</mml:mi> </mml:msup> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi mathvariant="normal"> Ω </mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">{H^k}(\Omega )</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and analytic solutions to selfadjoint elliptic problems. The polynomial implementation of this algorithm can be viewed as a nonconforming version of the <italic>h-p</italic> finite element method that also can produce the continuous approximations of the <italic>h-p</italic> method. The examples provided by our experiments provide discontinuous approximations that have errors similar to the finite element results.