A class of finite volume schemes of arbitrary order on nonuniform meshes1. Issue 5 (23rd December 2013)

• Record Type: Journal Article
• Authors: Zhang, Qinghui; Zou, Qingsong
• Is Part Of: Numerical methods for partial differential equations. Volume 30:Issue 5(2014:Sep.)
• Page Start: 1614
• Page End: 1632
• Publication Date: 2013-12-23
• Subjects: Differential equations, Partial -- Numerical solutions -- Periodicals; 515.353
• DOI: 10.1002/num.21853
• Languages: English
• ISSNs: 0749-159X

Abstract

<abstract abstract-type="main"> <title>Abstract</title> <p>In this article, we generalize the bi‐<inline-formula><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pghvbtpd0d" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley::media:num21853:num21853-math-0001" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>k</mml:mi><mml:mo>, </mml:mo><mml:mi>k</mml:mi><mml:mo>≥</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:math></alternatives></inline-formula> finite volume schemes developed in Zhang and Zou, J Sci Comput (in press) for elliptic equations with high smooth solutions to elliptic equations with singular solutions. By designing a special nonuniform rectangular meshes, we construct a class of finite volume schemes of arbitrary order <italic>k</italic>. Our theoretic analysis shows that if the solution has weak singularity of type <inline-formula><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pghvbtpczd" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley::media:num21853:num21853-math-0002" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msup><mml:mi>r</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo>|</mml:mo><mml:mi>log</mml:mi><mml:mi>r</mml:mi><mml:mo>|</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>, where <italic>r</italic> is the distance from some target point to some fixed singular point, the <italic>H</italic><sup>1</sup> norm of our finite volume schemes' discretization error converges with optimal order <inline-formula><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pghvbtpd2f" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley::media:num21853:num21853-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>N</mml:mi><mml:mrow><mml:mo>−</mml:mo><mml:mi>k</mml:mi></mml:mrow></mml:msup><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>, while the <italic>L</italic><sup>2</sup> norm error converges with order <inline-formula><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pghvbtpd1x" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley::media:num21853:num21853-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>N</mml:mi><mml:mrow><mml:mo>−</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:mi>k</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:math></alternatives></inline-formula>. Here, <italic>N</italic><sup>2</sup> is the cardinality of the underlying mesh. Superconvergence property of the scheme is also discussed. Our theoretic findings have been verified by numerical experiments. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 1614–1632, 2014</p> </abstract>