A hybridizable discontinuous Galerkin (HDG) method is introduced and analyzed to solve the Sobolev equation. The analysis includes the derivation of a priori error estimates, demonstrating that the approximations for both the flux and displacement exhibit convergence at a rate of order <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40314_2025_3364_Article_IEq1.gif" Format="GIF" Height="21" Rendition="HTML" Resolution="72" Type="Linedraw" Width="64" /> </InlineMediaObject> <EquationSource Format="TEX">$O(h^{k+1}),$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>O</mi> <mo stretchy="false">(</mo> <msup> <mi>h</mi> <mrow> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mo>,</mo> </mrow> </math></EquationSource> </InlineEquation> where h represents the mesh size and k is the polynomial degree. Additionally, the solution is further improved by applying a post-processing technique, and it has been demonstrated that, for <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40314_2025_3364_Article_IEq2.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="42" /> </InlineMediaObject> <EquationSource Format="TEX">$k\geqslant 1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>k</mi> <mo>⩾</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>, the post-processed solution converges at an enhanced rate of order <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40314_2025_3364_Article_IEq3.gif" Format="GIF" Height="21" Rendition="HTML" Resolution="72" Type="Linedraw" Width="59" /> </InlineMediaObject> <EquationSource Format="TEX">$O(h^{k+2})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>O</mi> <mo stretchy="false">(</mo> <msup> <mi>h</mi> <mrow> <mi>k</mi> <mo>+</mo> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>. A fully discrete scheme is also proposed, achieving second-order accuracy in time; numerical results are needed to validate the theoretical results.

A priori error estimates for sobolev equation using HDG method

期刊论文
发表日期：

2025年8月19日

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Shipra Gupta,
Sangita Yadav

摘要

A hybridizable discontinuous Galerkin (HDG) method is introduced and analyzed to solve the Sobolev equation. The analysis includes the derivation of a priori error estimates, demonstrating that the approximations for both the flux and displacement exhibit convergence at a rate of order $O(h^{k+1}),$ $O (h^{k + 1}),$ where h represents the mesh size and k is the polynomial degree. Additionally, the solution is further improved by applying a post-processing technique, and it has been demonstrated that, for $k\geqslant 1$ $k ⩾ 1$ , the post-processed solution converges at an enhanced rate of order $O(h^{k+2})$ $O (h^{k + 2})$ . A fully discrete scheme is also proposed, achieving second-order accuracy in time; numerical results are needed to validate the theoretical results.