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<p>In this article, we propose and analyze a least-squares-based weak Galerkin finite-element method (WG-FEM) for solving the indefinite time-harmonic Maxwell’s equations in <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\({\mathbb {R}}^d\;(d=2, 3).\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>d</mi> </msup> <mspace width="0.277778em" /> <mrow> <mo stretchy="false">(</mo> <mi>d</mi> <mo>=</mo> <mn>2</mn> <mo>,</mo> <mn>3</mn> <mo stretchy="false">)</mo> </mrow> <mo>.</mo> </mrow> </math></EquationSource> </InlineEquation> The super-convergence of order one for the discrete <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\textbf{H}^1\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi mathvariant="bold">H</mi> <mn>1</mn> </msup> </math></EquationSource> </InlineEquation>-like norm has been established. Numerical simulations show that the approximate solutions converge to the exact solutions with optimal rates in the <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(L^2\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mn>2</mn> </msup> </math></EquationSource> </InlineEquation> norm on hybrid meshes. In addition, this method is shown to be absolutely stable under low regularity requirements with high wave numbers.</p>

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A Least-Squares-Based Weak Galerkin Finite-Element Method for the Time-Harmonic Maxwell’s Equations

  • 期刊论文

  • 发表日期：

    2025年8月31日

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  • Raman Kumar,
  • Bhupen Deka

摘要

In this article, we propose and analyze a least-squares-based weak Galerkin finite-element method (WG-FEM) for solving the indefinite time-harmonic Maxwell’s equations in \({\mathbb {R}}^d\;(d=2, 3).\) R d ( d = 2 , 3 ) . The super-convergence of order one for the discrete \(\textbf{H}^1\) H 1 -like norm has been established. Numerical simulations show that the approximate solutions converge to the exact solutions with optimal rates in the \(L^2\) L 2 norm on hybrid meshes. In addition, this method is shown to be absolutely stable under low regularity requirements with high wave numbers.