Higher order local accuracy by averaging in the finite element method

Abstract

Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="u Subscript h"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>u</mml:mi> <mml:mi>h</mml:mi> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">{u_h}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be a Ritz-Galerkin approximation, corresponding to the solution <italic>u</italic> of an elliptic boundary value problem, which is based on a uniform subdivision in the interior of the domain. In this paper we show that by "averaging" the values of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="u Subscript h"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>u</mml:mi> <mml:mi>h</mml:mi> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">{u_h}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> in the neighborhood of a point <italic>x</italic> we may (for a wide class of problems) construct an approximation to <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="u left-parenthesis x right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>u</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">u(x)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> which is often a better approximation than <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="u Subscript h Baseline left-parenthesis x right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>u</mml:mi> <mml:mi>h</mml:mi> </mml:msub> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">{u_h}(x)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> itself. The "averaging" operator does not depend on the specific elliptic operator involved and is easily constructed.