Superconvergence of the local discontinuous galerkin method applied to the one‐dimensional second‐order wave equation. Issue 3 (23rd December 2013)

• Record Type: Journal Article
• Authors: Baccouch, Mahboub
• Is Part Of: Numerical methods for partial differential equations. Volume 30:Issue 3(2014:May)
• Page Start: 862
• Page End: 901
• Publication Date: 2013-12-23
• Subjects: Differential equations, Partial -- Numerical solutions -- Periodicals; 515.353
• DOI: 10.1002/num.21840
• Languages: English
• ISSNs: 0749-159X

Abstract

<abstract abstract-type="main"> <title> <x xml:space="preserve">Abstract</x> </title> <p>We analyze the superconvergence properties of the local discontinuous Galerkin (LDG) method applied to the second‐order wave equation in one space dimension. With a suitable projection of the initial conditions for the LDG scheme, we prove that the LDG solution and its spatial derivative are <alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgg4vp2kp37" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:0749159X:media:num21840:num21840-math-0001" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi mathvariant="script">O</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:msup><mml:mi>h</mml:mi><mml:mrow><mml:mi>p</mml:mi><mml:mo>+</mml:mo><mml:mn>3</mml:mn><mml:mo>/</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:math></alternatives> super close to particular projections of the exact solutions for <italic>pth</italic>‐degree polynomial spaces. We use these results to show that the significant parts of the discretization errors for the LDG solution and its derivative are proportional to <alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgg4vp2kp14" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:0749159X:media:num21840:num21840-math-0002" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>p</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:math></alternatives>‐degree right and left Radau polynomials, respectively. These results allow us to prove that the <italic>p</italic>‐degree LDG solution and its derivative are <alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgg4vp2kq03" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:0749159X:media:num21840:num21840-math-0003" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi mathvariant="script">O</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:msup><mml:mi>h</mml:mi><mml:mrow><mml:mi>p</mml:mi><mml:mo>+</mml:mo><mml:mn>3</mml:mn><mml:mo>/</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:math></alternatives> superconvergent at the roots of <alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgg4vp2kq1n" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:0749159X:media:num21840:num21840-math-0004" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>p</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:math></alternatives>‐degree right and left Radau polynomials, respectively, while computational results show higher <alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgg4vp2kpxg" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:0749159X:media:num21840:num21840-math-0005" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi mathvariant="script">O</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:msup><mml:mi>h</mml:mi><mml:mrow><mml:mi>p</mml:mi><mml:mo>+</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:math></alternatives> convergence rate. Superconvergence results can be used to construct asymptotically correct <italic>a posteriori</italic> error estimates by solving a local steady problem on each element. This will be discussed further in Part II of this work, where we will prove that the <italic>a posteriori</italic> LDG error estimates for the solution and its derivative converge to the true spatial errors in the <italic>L</italic><sup>2</sup>‐norm under mesh refinement. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 862–901, 2014</p> </abstract>