The resulting V-cycle, used as a conjugate-gradient preconditioner for continuous lagrangian finite element discretizations, maintains stable iteration counts and remains effective in regimes where standard AMG deteriorates.
Abstract
Algebraic multigrid (AMG) methods are robust and efficient black-box preconditioners for linear systems arising from low-order discretizations of elliptic partial differential equations, but their performance often deteriorates for high-order methods. We introduce an algebraic-geometric multilevel preconditioner for continuous lagrangian finite element discretizations. The method automatically constructs prolongation operators and a Galerkin hierarchy using only the coordinates of finite element support points, without requiring a prescribed mesh hierarchy or domain decomposition. The support-point are recursively partitioned by an R-tree algorithm based on axis-aligned bounding boxes, producing a hierarchy of agglomerates. In contrast to classical AMG, which typically uses piecewise-constant aggregate coarse spaces, our method embeds local discontinuous polynomial spaces of degree $p'>0$, defined on the agglomerate boxes, through continuous nodal interpolation. This yields a high-order extension of the Nicolaides coarse space. A two-level analysis quantifies how the coarse polynomial degree offsets the effects of large agglomerates and high-order fine discretizations, under uniform box-regularity, interpolation-stability, and stable-decomposition assumptions. Numerical experiments in two and three dimensions show that the resulting V-cycle, used as a conjugate-gradient preconditioner, maintains stable iteration counts and remains effective in regimes where standard AMG deteriorates.
The solution of large-scale symmetric positive definite linear systems arising from discretizations of second-order elliptic equations is challenging, especially in applications with highly heterogeneous coefficients, such as flow in porous media, which can lead to severely ill-conditioned systems. In this context, multiscale methods have recently been used to accelerate Krylov subspace methods, owing to their favorable parallel scalability. In this work, we present AlgMortar, a fully algebraic realization of the Multiscale Mortar Mixed Finite Element Method (MMMFEM). The method uses only information extracted from the fine-grid system matrix, which facilitates its implementation in existing solvers. AlgMortar uses graph partitioning to define a domain decomposition directly from the matrix graph. On each subdomain, it builds local linear systems that mimic Dirichlet problems, and couples the resulting local solutions through an algebraic interface condition that recovers the weak flux-continuity mechanism of MMMFEM. We prove that the method is well posed when the fine-grid matrix is symmetric positive definite and has nonpositive off-diagonal entries, a structure commonly arising from discretizations of elliptic problems. Numerical experiments on fine-grid linear systems arising from finite-volume discretizations of Darcy flow problems show that, when used as a preconditioner for the conjugate gradient method, the proposed approach exhibits good scalability and is competitive with state-of-the-art algebraic multigrid methods for challenging heterogeneous, high-contrast test cases, including highly irregular corner-point grids.
Luan F. Santos, F. S. Sousa, R. Ausas et al.· arXiv.org· 0 citations
It is proved that, if the BJ, BGS, and MS preconditioners are GLT sequences with symbol $\kappa$, then the sequences of the BJ, BGS, and MS preconditioners are GLT sequences with symbol $\kappa$.
C. Garoni, Abdessadek Rifqui, S. Serra-Capizzano· 0 citations
In this work, we present two novel strategies to impose high-order boundary conditions on fixed and moving curved domains, approximated with piecewise affine triangulations. Achieving high-order accuracy on curved domains requires tackling both the PDE discretization error and the geometrical error simultaneously. While the former can be reduced by employing high-order numerical methods such as finite volume and discontinuous Galerkin, the latter demands either a high-order parametrization of the physical domain or a consistent approximation of the boundary conditions. Minimization-based approaches like the Reconstruction for Off-site Data (ROD) method allow one to skip the construction of high-order curvilinear meshes by defining high-order consistent boundary conditions on a computational boundary that does not match the physical one. The ROD approach mitigates the second-order geometrical error by retrieving a modified polynomial in each boundary cell, which enforces the boundary conditions exactly on the physical boundary. However, the standard ROD method requires the inversion of a local linear system, whose cost grows with the polynomial degree and mesh refinement. Inspired by a recent one-dimensional analysis, we show that ROD-type approaches can be recast as simple polynomial corrections, applicable without any linear system inversion. This greatly simplifies the development of minimization-based boundary treatments and reduces the associated computational cost. To prove the wide applicability of our strategy, we develop it within a Runge-Kutta discontinuous Galerkin framework and an ADER arbitrary-Lagrangian-Eulerian finite volume framework for compressible flows on fixed and moving domains. Several numerical experiments with Dirichlet and slip-wall boundary conditions are presented, with convergence analysis up to fifth order in both 2D and 3D.
We propose an adaptive multigrid (MG) method for discontinuous Galerkin formulations of elliptic problems using Brandt's full approximation scheme (FAS). Unlike common approaches, this method achieves local $hp$-refinement of hexahedral meshes without the need for hanging nodes. The core component of the FAS-MG method is an overlapping Schwarz smoother, which is optionally accelerated by a Krylov method. This smoother is designed for unstructured curvilinear meshes but maintains a tensor-product structure for fast diagonalization. Numerical experiments demonstrate the exceptional efficiency of the FAS-MG method. Dedicated studies confirm its robustness against high aspect ratios, element deformation, and irregular mesh topology. We also verify its capability for dynamic parallel mesh adaptation using the wave-front benchmark of \v{C}erven\'y, Dobrev, and Kolev (SIAM J. Sci. Comp. 41, 2019). Finally, we present preliminary results of extending the method to incompressible Navier-Stokes problems.
Multi-dimensional uniform-positivity-preserving (UPP) methods are proposed for high-order finite-volume multi-resolution weighted essentially non-oscillatory schemes with adaptive linear weights (ALW-MR-WENO schemes) when solving the Euler equations on structured meshes. By utilizing two central spatial stencils, the ALW-MR-WENO schemes generate a polynomial vector and maintain high-order accuracy in smooth domains while preserving the non-oscillatory property near strong discontinuities. The cell averages are reformulated to construct the auxiliary polynomial vectors for the UPP methods. By applying a nested bisection method, the positivity of the density polynomial and the polynomial on the numerator of pressure rational function are detected over the computational cell instead of at some Gauss–Lobatto quadrature points as usual. Once the negativity occurs, a novel nonlinear compression limiter is imposed to suppress the positivity of density and pressure functions throughout the computational cell while preserving the high-order accuracy. Furthermore, this work introduces the Hu–Tan–Zhu formulas for algebraic polynomials of corresponding degrees that require only three points and ensure a larger boundary quadrature weight of 1/6, where all quadrature weights remain positive and their summation is one. These new formulas enable a rigorous proof of uniformly sufficient Courant–Friedrichs–Lewy number of 1/6 for arbitrary high-order UPP ALW-MR-WENO schemes, marking a significant improvement over the classical positivity-preserving (PP) methods. The numerical results indicate that the UPP methods reduce approximately 5%–65% computational time compared to the classical PP methods with their sufficient Courant–Friedrichs–Lewy numbers are 1/12, 1/20, or 1/30 as the ALW-MR-WENO schemes increase their precisions from the fifth- to seventh- or ninth-order accuracy, respectively.
Peng-le Hu, Tan Yan, Jun Zhu· The Physics of Fluids· 0 citations
We present a high-order radial basis function-generated finite difference (RBF-FD) method for partial differential equations on moving manifolds $\mathcal M(t)\subset\mathbb R^3$ of co-dimension one. Our method builds on the tangent-plane formulation of surface RBF-FD and combines Lagrangian and Eulerian treatments: the manifold and material derivative are evolved in a Lagrangian fashion, while the remaining surface differential operators are reconstructed on the instantaneous point cloud and stabilized, when necessary, by a quasi-analytical hyperviscosity formulation. Lagrangian marker drift is handled by adaptive rearrangement using a compact global parametric model of the moving surface, which also supplies accurate normals and geometry-based quadrature. After rearrangement, we reconstruct the multistep history by backward semi-Lagrangian tracing and interpolation before resuming the Lagrangian time discretization. We exploit temporal coherence through three update strategies: defect correction for the local RBF-FD weights, a curvature-based update of the hyperviscosity coefficients between spectral recomputations, and a global defect-correction iteration that reuses an incomplete LU (ILU) factorization before preconditioned generalized minimal residual (GMRES) iterations. Finally, we enforce the prescribed global mass balance through a scalar projection based on the evolving surface quadrature. Numerical experiments demonstrate high-order convergence, conservation to roundoff in source-free problems, stable long-time integration, and substantial savings from the proposed update strategies.
Matthew Lowery, G. Wright, Varun Shankar· 0 citations
We use cookies to run the site and, with your consent, for analytics and to show ads.
See our Cookie Policy.
