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Mechanistic models play a crucial role in algorithms for numerical problem-solving, particularly in nonlinear mixed effects modeling (NMEM). These models aim to minimize specific objective functions by evaluating various parameter estimates, leading to the development of systematic algorithms. In some cases, linearization techniques approximate the model using linear equations.
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A Sparse Hierarchical hp-Finite Element Method on Disks and Annuli.

Ioannis P A Papadopoulos1, Sheehan Olver2

  • 1Weierstrass Institute for Applied Analysis and Stochastics, Berlin, Germany.

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|June 26, 2025
PubMed
Summary

A new sparse hierarchical hp-finite element method (hp-FEM) efficiently solves the Helmholtz equation on disk and annulus domains. This method handles complex coefficients and discontinuities, enabling faster computations.

Keywords:
AnnulusCylinderDiskHigh-frequency Helmholtz equationQuasi-optimal complexitySchrödinger equationhp-finite element method

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Area of Science:

  • Computational Mathematics
  • Numerical Analysis
  • Partial Differential Equations

Background:

  • The Helmholtz equation is crucial in wave propagation problems.
  • Solving Helmholtz equations with variable coefficients and complex geometries is computationally demanding.
  • Existing methods often struggle with discontinuities and anisotropic properties.

Purpose of the Study:

  • To develop an efficient and robust numerical method for the Helmholtz equation.
  • To handle variable coefficients, radial discontinuities, and anisotropic properties.
  • To achieve optimal complexity factorization and quasi-optimal solvers.

Main Methods:

  • Sparse hierarchical hp-finite element method (hp-FEM) on disk/annulus domains.
  • Preservation of Fourier mode decoupling for rotationally invariant operators.
  • Block diagonal mass and stiffness matrices with optimal complexity factorization.
  • Handling of radial discontinuities and anisotropic coefficients.

Main Results:

  • The hp-FEM yields sparse matrices with a pattern independent of discretization order.
  • The method effectively handles radial discontinuities in source terms and coefficients.
  • Rotationally anisotropic coefficients approximated by polynomials also lead to sparse systems.
  • Quasi-optimal solutions are achieved using the Alternating Direction Implicit (ADI) algorithm.

Conclusions:

  • The developed sparse hp-FEM offers an efficient solution for the Helmholtz equation.
  • The method demonstrates robustness in handling challenging coefficient variations and discontinuities.
  • The approach is extendable to 3D domains and related equations like the Schrödinger equation.