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<p>Singularly perturbed problems arise in several applications and are known to contain boundary layers which are rapidly varying solution components due to the presence of a parameter which is usually very small in practice. In this work, we present least-squares spectral element methods for one dimensional elliptic boundary layer problems, employing both boundary layer (<i>rp</i> type) and geometric (<i>hp</i> type) mesh refinements. Stability estimates are derived and a numerical scheme, based on minimizing residuals in the least-squares sense with respect to suitable Sobolev norms is presented. We also design preconditioners that are essentially modified, parameter dependent <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(H^2\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>H</mi> <mn>2</mn> </msup> </math></EquationSource> </InlineEquation>-norms with respect to the type of discretization used in decomposition. The proposed method achieves exponential convergence rates of the form <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(O\left( e^{-bW}\right) \)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>O</mi> <mfenced close=")" open="("> <msup> <mi>e</mi> <mrow> <mo>-</mo> <mi>b</mi> <mi>W</mi> </mrow> </msup> </mfenced> </mrow> </math></EquationSource> </InlineEquation>, where <i>W</i> denotes the polynomial order and the error in the approximation is independent of the boundary layer parameter <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(\epsilon \)</EquationSource> <EquationSource Format="MATHML"><math> <mi>ϵ</mi> </math></EquationSource> </InlineEquation>. Furthermore, the computational complexity of the method is discussed and numerical results are provided to confirm the theoretical results and validate the error estimates and computational complexity of the proposed approach.</p>

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Exponentially accurate least-squares spectral element method for 1D elliptic boundary layer problems

  • 期刊论文

  • 发表日期：

    2026年1月5日

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  • Akhlaq Husain,
  • Aliya Kazmi,
  • Subhashree Mohapatra,
  • Ziya Uddin,
  • Mashael I. Alammari

摘要

Singularly perturbed problems arise in several applications and are known to contain boundary layers which are rapidly varying solution components due to the presence of a parameter which is usually very small in practice. In this work, we present least-squares spectral element methods for one dimensional elliptic boundary layer problems, employing both boundary layer (rp type) and geometric (hp type) mesh refinements. Stability estimates are derived and a numerical scheme, based on minimizing residuals in the least-squares sense with respect to suitable Sobolev norms is presented. We also design preconditioners that are essentially modified, parameter dependent \(H^2\) H 2 -norms with respect to the type of discretization used in decomposition. The proposed method achieves exponential convergence rates of the form \(O\left( e^{-bW}\right) \) O e - b W , where W denotes the polynomial order and the error in the approximation is independent of the boundary layer parameter \(\epsilon \) ϵ . Furthermore, the computational complexity of the method is discussed and numerical results are provided to confirm the theoretical results and validate the error estimates and computational complexity of the proposed approach.