<p>We introduce and analyze a hybridizable discontinuous Galerkin (HDG) approach for the strongly damped linear wave equation. In our investigation, we derive a priori error estimates to demonstrate the optimal convergence of the approximations for both the solution and its gradient. Further, with the help of the dual problem, we present a post-processed solution and analyze its convergence rate, which is of order <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10915_2024_2762_Article_IEq1.gif" Format="GIF" Height="15" Rendition="HTML" Resolution="72" Type="Linedraw" Width="39" /> </InlineMediaObject> <EquationSource Format="TEX">$k+2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>k</mi> <mo>+</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation> for <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10915_2024_2762_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>, where <i>k</i> is the degree of the polynomial. We also propose a fully discrete scheme, which is of <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10915_2024_2762_Article_IEq3.gif" Format="GIF" Height="21" Rendition="HTML" Resolution="72" Type="Linedraw" Width="117" /> </InlineMediaObject> <EquationSource Format="TEX">$O((\Delta t)^2+h^{k+1})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>O</mi> <mo stretchy="false">(</mo> <msup> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="normal">Δ</mi> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mn>2</mn> </msup> <mo>+</mo> <msup> <mi>h</mi> <mrow> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>. To validate our theoretical findings, we perform numerical experiments.</p>

Hybridizable Discontinuous Galerkin Method for Strongly Damped Wave Problem

期刊论文
发表日期：

2025年1月20日

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Shipra Gupta,
Sangita Yadav

摘要

We introduce and analyze a hybridizable discontinuous Galerkin (HDG) approach for the strongly damped linear wave equation. In our investigation, we derive a priori error estimates to demonstrate the optimal convergence of the approximations for both the solution and its gradient. Further, with the help of the dual problem, we present a post-processed solution and analyze its convergence rate, which is of order $k+2$ $k + 2$ for $k \geqslant 1$ $k ⩾ 1$ , where k is the degree of the polynomial. We also propose a fully discrete scheme, which is of $O((\Delta t)^2+h^{k+1})$ $O ({(Δ t)}^{2} + h^{k + 1})$ . To validate our theoretical findings, we perform numerical experiments.