A new error analysis for discontinuous finite element methods for linear elliptic problems

Abstract

The standard <italic>a priori</italic> error analysis of discontinuous Galerkin methods requires additional regularity on the solution of the elliptic boundary value problem in order to justify the Galerkin orthogonality and to handle the normal derivative on element interfaces that appear in the discrete energy norm. In this paper, a new error analysis of discontinuous Galerkin methods is developed using only the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper H Superscript k"> <mml:semantics> <mml:msup> <mml:mi>H</mml:mi> <mml:mi>k</mml:mi> </mml:msup> <mml:annotation encoding="application/x-tex">H^k</mml:annotation> </mml:semantics> </mml:math> </inline-formula> weak formulation of a boundary value problem of order <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="2 k"> <mml:semantics> <mml:mrow> <mml:mn>2</mml:mn> <mml:mi>k</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">2k</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . This is accomplished by replacing the Galerkin orthogonality with estimates borrowed from <italic>a posteriori</italic> error analysis and by using a discrete energy norm that is well defined for functions in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper H Superscript k"> <mml:semantics> <mml:msup> <mml:mi>H</mml:mi> <mml:mi>k</mml:mi> </mml:msup> <mml:annotation encoding="application/x-tex">H^k</mml:annotation> </mml:semantics> </mml:math> </inline-formula> .