## Abstract

In this paper, we developed and demonstrated a non-nested multilevel procedure for solving the heat conduction equation in complex domains using a meshless discretization method. Previous multilevel methods for solving elliptic partial differential equations in complex domains have mostly used one of the four approaches: nested refinement, agglomeration, additive corrections, and algebraic coarsening. Each of these techniques has some issues of generality, robustness, and speed of acceleration. In this paper, we developed a generally applicable multilevel algorithm for partial differential equations discretized on complex domains using unstructured finite volume, finite element, and meshless methods. We applied this multilevel method to accelerate convergence of the set of discrete equations obtained by a meshless technique. The heat conduction equation is discretized at scattered points using a polyharmonic spine (PHS) radial basis function (RBF) interpolation with appended polynomials to achieve exponential convergence of discretization errors. The RBF interpolations are performed over clouds of points, and the partial differential equation is collocated at the scattered points. The multilevel algorithm to solve the set of linear equations utilizes multiple independently generated coarser sets of points. Restriction of residuals and prolongation of the corrections are also performed using the RBF interpolations. The fast convergence of the algorithm is demonstrated for solution of the heat conduction equation in three model complex domains with manufactured solutions. A simple successive over-relaxation point solver is used as the relaxation scheme.

Original language | English (US) |
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Pages (from-to) | 55-70 |

Number of pages | 16 |

Journal | Computational Thermal Sciences |

Volume | 16 |

Issue number | 3 |

DOIs | |

State | Published - 2024 |

Externally published | Yes |

## Keywords

- heat conduction equation
- meshless discretization
- non-nested multilevel method
- polyharmonic splines
- radial basis functions

## ASJC Scopus subject areas

- Energy Engineering and Power Technology
- Surfaces and Interfaces
- Fluid Flow and Transfer Processes
- Computational Mathematics