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<p>In this work, a modified numerical scheme is developed based on the scale-3 Haar wavelet for solving elliptic partial differential equations (PDEs) that describe the Helmholtz equation and 3D Poisson equation. The spatial derivatives are discretized scale-3 Haar wavelet expansions, which are then integrated and extended to a two and three dimensional solution via Kronecker tensor product, incorporating boundary conditions through integration constants. Furthermore, theoretical convergence analysis of the proposed method is discussed and supported with numerical evaluation of maximum absolute errors (<InlineEquation ID="IEq1"><EquationSource Format="MATHML"><math><msub><mi>L</mi><mi mathvariant="normal">∞</mi></msub></math></EquationSource><EquationSource Format="TEX">$L_{\infty}$</EquationSource></InlineEquation>), mean squared errors (<InlineEquation ID="IEq2"><EquationSource Format="MATHML"><math><msub><mi>L</mi><mn>2</mn></msub></math></EquationSource><EquationSource Format="TEX">$L_{2}$</EquationSource></InlineEquation>), and the computational convergence rate across resolution levels, validating the method’s convergence. Computational simulations are executed using MATLAB programming. The wavelet method is compared with the existing finite difference method and scale-2 Haar wavelet method, and the results demonstrate that while all three approaches effectively solve elliptic PDEs, the scale-3 Haar wavelet method outperforms the others by delivering more accurate approximate solutions with greater efficiency. The results of this investigation establish the potential and reliability of Haar wavelet methods for solving intricate PDEs in various engineering domains.</p>

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Convergence analysis of a modified scale-3 Haar wavelet method for the numerical solution of the Helmholtz equation

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2026年7月20日

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  • Avinash K,
  • Harinakshi Karkera

摘要

In this work, a modified numerical scheme is developed based on the scale-3 Haar wavelet for solving elliptic partial differential equations (PDEs) that describe the Helmholtz equation and 3D Poisson equation. The spatial derivatives are discretized scale-3 Haar wavelet expansions, which are then integrated and extended to a two and three dimensional solution via Kronecker tensor product, incorporating boundary conditions through integration constants. Furthermore, theoretical convergence analysis of the proposed method is discussed and supported with numerical evaluation of maximum absolute errors (L∞$L_{\infty}$), mean squared errors (L2$L_{2}$), and the computational convergence rate across resolution levels, validating the method’s convergence. Computational simulations are executed using MATLAB programming. The wavelet method is compared with the existing finite difference method and scale-2 Haar wavelet method, and the results demonstrate that while all three approaches effectively solve elliptic PDEs, the scale-3 Haar wavelet method outperforms the others by delivering more accurate approximate solutions with greater efficiency. The results of this investigation establish the potential and reliability of Haar wavelet methods for solving intricate PDEs in various engineering domains.