Asymptotically exact a posteriori local discontinuous Galerkin error estimates for the one‐dimensional second‐order wave equation. Issue 5 (23rd December 2014)

• Record Type: Journal Article
• Authors: Baccouch, Mahboub
• Is Part Of: Numerical methods for partial differential equations. Volume 31:Issue 5(2015)
• Page Start: 1461
• Page End: 1491
• Publication Date: 2014-12-23
• Subjects: Differential equations, Partial -- Numerical solutions -- Periodicals; 515.353
• DOI: 10.1002/num.21955
• Languages: English
• ISSNs: 0749-159X

Abstract

<abstract abstract-type="main"> <title> <x xml:space="preserve">Abstract</x> </title> <p>In this article, we analyze a residual‐based a posteriori error estimates of the spatial errors for the semidiscrete local discontinuous Galerkin (LDG) method applied to the one‐dimensional second‐order wave equation. These error estimates are computationally simple and are obtained by solving a local steady problem with no boundary condition on each element. We apply the optimal <italic>L</italic><sup>2</sup> error estimates and the superconvergence results of Part I of this work [Baccouch, Numer Methods Partial Differential Equations 30 (2014), 862–901] to prove that, for smooth solutions, these a posteriori LDG error estimates for the solution and its spatial derivative, at a fixed time, converge to the true spatial errors in the <italic>L</italic><sup>2</sup>‐norm under mesh refinement. The order of convergence is proved to be <inline-formula><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgj25zbt6zd" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:0749159X:media:num21955:num21955-math-0001" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><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:math></alternatives></inline-formula>, when <italic>p</italic>‐degree piecewise polynomials with <inline-formula><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgj25zbt734" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:0749159X:media:num21955:num21955-math-0002" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>p</mml:mi><mml:mo>≥</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:math></alternatives></inline-formula> are used. As a consequence, we prove that the LDG method combined with the a posteriori error estimation procedure yields both accurate error estimates and <inline-formula><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgj25zbt72k" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:0749159X:media:num21955:num21955-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></inline-formula> superconvergent solutions. Our computational results show higher <inline-formula><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgj25zbt78w" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:0749159X:media:num21955:num21955-math-0004" 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></inline-formula> convergence rate. We further prove that the global effectivity indices, for both the solution and its derivative, in the <italic>L</italic><sup>2</sup>‐norm converge to unity at <inline-formula><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgj25zbt757" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:0749159X:media:num21955:num21955-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:mn>1</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></inline-formula> rate while numerically they exhibit <inline-formula><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgj25zbt7fn" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:0749159X:media:num21955:num21955-math-0006" 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:mn>2</mml:mn></mml:msup><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula> and <inline-formula><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgj25zbt7b0" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:0749159X:media:num21955:num21955-math-0007" 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:mi>h</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula> rates, respectively. Numerical experiments are shown to validate the theoretical results. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 1461–1491, 2015</p> </abstract>