arXiv subjects

Jordi Manyer

Publications and source records attributed to Jordi Manyer.

8 recordsLinked to original sources

arXiv2026

Geometric Multigrid Preconditioners for conforming Finite Element Methods on Arbitrarily Agglomerated Meshes

Geometric multigrid methods for conforming finite element discretisations classically require a nested mesh hierarchy, forcing coarse meshes to be conforming, which is unsuitable for general domains. In order to eliminate this geometric constraint, we introduce and analyse a family of multilevel auxiliary space preconditioners. At the first level, a conforming Lagrangian discretisation on a fine simplicial mesh is preconditioned by a non-conforming polytopal discretisation posed on an arbitrarily agglomerated coarse mesh. At the subsequent levels, the coarse problem is solved by a geometric multigrid method for that polytopal discretisation on further agglomerated meshes, so that the whole hierarchy is generated from the fine mesh by agglomeration alone. We consider two polytopal discretisations, a symmetric interior penalty discontinuous Galerkin method and a hybrid high-order method. The prolongation operator at the first level is an element-wise interpolation onto the fine mesh followed by nodal averaging, preceded in the hybrid high-order case by a local potential reconstruction. For point-Jacobi and vertex-star patch additive Schwarz smoothers, we prove that the condition number of the preconditioned system is bounded by $O((H/h)^3)$, where $H/h$ is the coarsening ratio, and by $O((H/h)^2)$ for the DG variant with penalty proportional to $H/h$. Numerical experiments in two and three dimensions, on structured and unstructured meshes with agglomerates generated by automatic clustering algorithms, confirm the robustness of the preconditioners.

math.NA↗

arXiv2026

Geometric multigrid preconditioners for unfitted discretisations of the de Rham complex

We propose two families of multigrid preconditioners for unfitted finite element discretisations of the Riesz maps of the de Rham complex, in H(grad), H(curl) and H(div), stabilised by cell aggregation. The first is a geometric multigrid method applied directly to the unfitted operators, on a hierarchy of background meshes that share the same boundary representation. Its smoothers are star patches built on the aggregated mesh, so that each patch captures the kernel of the stabilisation. The second is a multilevel auxiliary-space preconditioner that combines a patch smoother localised around the cut cells with the body-fitted problem on the active mesh, which is in turn solved by a non-nested geometric multigrid method. Both are used as building blocks of block-diagonal preconditioners for the mixed Hodge Laplacians. Numerical experiments in two and three dimensions, on geometries with a controlled cut fraction and on general geometries, show iteration counts that are robust with respect to the cut configuration, the mesh size and the number of levels. We also report results for higher-order discretisations, weakly imposed Dirichlet boundary conditions and a face-based stabilisation.

math.NA↗

arXiv2026

Rotating bases for finite element exterior calculus: closed-form change of basis under vertex permutations

The geometrically decomposed bases of the polynomial differential form spaces $\mathcal{P}_rΛ^1$ and $\mathcal{P}_r^-Λ^1$ on a simplex are products of a barycentric scalar polynomial and a directional or Whitney 1-form. These bases depend on an ordering of the simplex vertices. On unstructured meshes, neighbouring cells need not agree on this ordering, making it non-trivial to achieve conformity in finite element software. Existing strategies compute degree-of-freedom transformation matrices numerically, or impose a global vertex ordering through face-bubble spaces or by preprocessing the mesh. We consider the standard basis of the $\mathcal{P}_r^-Λ^1$ (trimmed) space and introduce a new basis for the $\mathcal{P}_rΛ^1$ (full) space, for arbitrary polynomial order $r \geq 1$ and dimension $D \geq 2$, and prescribe their basis polynomials as shape functions. We show that the pullback of shape functions under the change of coordinates induced by an arbitrary vertex relabelling $π$ in the symmetric group $S_{D+1}$ admits a closed-form, combinatorial expression. For most basis functions this pullback is a single basis function of the relabelled ordering, up to sign. On an explicitly characterised "filter-hit" set, this pullback is a signed sum of at most $D$ (full space) or exactly two (trimmed space) relabelled basis functions. All coefficients are in $\{-1, +1\}$ for all $r$, $D$ and $π$. The inverse transformation is obtained by computing the formulas at $π^{-1}$, so no numerical inversion of a change-of-basis matrix is needed. Conforming assembly and evaluation on simplicial meshes with arbitrary vertex orderings reduce to index manipulation. Our results are compared with the study of relabelling-invariant bases of Berchenko-Kogan and Licht, and verified in an open-source Julia implementation in the Gridap.jl library.

math.NA↗

arXiv2026

Geometric Multigrid solvers for Hybrid High-Order methods on polytopal meshes

We propose the first optimal geometric multigrid solver for hybrid high-order discretisations that can handle arbitrary polytopal agglomeration hierarchies in both two and three dimensions. The key ingredient is the use of modified skeleton spaces, which naturally accommodate non-planar interfaces arising during coarsening while reducing the number of degrees of freedom. We prove robust convergence with respect to the mesh size and the number of levels, and we validate our results numerically on a range of agglomeration-based mesh hierarchies. The approach extends naturally to other hybrid discretisations such as hybridizable discontinuous Galerkin and Weak Galerkin methods.

math.NA↗

arXiv2026

A natural language framework for non-conforming hybrid polytopal methods in Gridap.jl

Hybrid finite element methods such as hybridizable discontinuous Galerkin, hybrid high-order and weak Galerkin have emerged as powerful techniques for solving partial differential equations on general polytopal meshes. Despite their diverse mathematical origins, these methods share a common computational structure involving hybrid discrete spaces, local projection operators and static condensation. This work presents a comprehensive framework for implementing such methods within the Gridap finite element library. We introduce new abstractions for polytopal mesh representation using graph-based structures, broken polynomial spaces on arbitrary mesh entities, patch-based local assembly for cell-wise linear systems, high-level local operator construction and automated static condensation. These abstractions enable concise implementations of hybrid methods while maintaining computational efficiency through Julia's just-in-time compilation and Gridap's lazy evaluation strategies. We demonstrate the framework through implementations of several non-conforming polytopal methods for the Poisson problem, linear elasticity, incompressible Stokes flow and optimal control on polytopal meshes.

cs.MS↗

arXiv2025

Analysis of BDDC preconditioners for non-conforming polytopal hybrid discretisation methods

In this work, we build on the discrete trace theory developed by Badia, Droniou, and Tushar (Foundations of Computational Mathematics, in press, 2025; \href{https://doi.org/10.1007/s10208-025-09734-6}{doi:10.1007/s10208-025-09734-6}) to analyze the convergence rate of the Balancing Domain Decomposition by Constraints (BDDC) preconditioner generated from non-conforming polytopal hybrid discretizations. We prove polylogarithmic bounds on the condition number for the preconditioner that are independent of the mesh parameter and the number of subdomains, and that hold on polytopal meshes. The analysis relies on the continuity of a face truncation operator, which we establish in the fully discrete polytopal setting. To validate the theory, we present numerical experiments that confirm the truncation estimate and condition number bounds. In particular, we conduct weak scalability tests for second-order elliptic problems discretized using discontinuous skeletal methods, specifically Hybridizable Discontinuous Galerkin (HDG) and Hybrid High-Order (HHO) methods. We also demonstrate the robustness of the preconditioner for piecewise discontinuous coefficients with large jumps.

math.NA↗

arXiv2025

Level-set topology optimisation with unfitted finite elements and automatic shape differentiation

In this paper we develop automatic shape differentiation techniques for unfitted discretisations and link these to recent advances in shape calculus for unfitted methods. We extend existing analytic shape calculus results to the case where the domain boundary intersects with the boundary of the background domain. We further show that we can recover these analytic derivatives to machine precision regardless of the mesh size using the developed automatic shape differentiation techniques, drastically reducing the burden associated with the analytic derivation of these quantities. In addition, we show that we can also recover the symmetric shape Hessian. We implement these techniques for both serial and distributed computing frameworks in the Julia package GridapTopOpt and the wider Gridap ecosystem. As part of this implementation we propose a novel graph-based approach for isolated volume detection. We demonstrate the applicability of the unfitted automatic shape differentiation framework and our implementation by considering the three-dimensional minimum compliance topology optimisation of a linear elastic wheel and of a linear elastic structure in a fluid-structure interaction problem with Stokes flow. The implementation is general and allows GridapTopOpt to solve a wider range of problems on unstructured meshes without analytic calculation of shape derivatives and avoiding issues that arise when material properties are smoothed at the domain boundary. The software is open source and available at https://github.com/zjwegert/GridapTopOpt.jl.

math.OC↗

arXiv2024

GridapTopOpt.jl: A scalable Julia toolbox for level set-based topology optimisation

In this paper we present GridapTopOpt, an extendable framework for level set-based topology optimisation that can be readily distributed across a personal computer or high-performance computing cluster. The package is written in Julia and uses the Gridap package ecosystem for parallel finite element assembly from arbitrary weak formulations of partial differential equation (PDEs) along with the scalable solvers from the Portable and Extendable Toolkit for Scientific Computing (PETSc). The resulting user interface is intuitive and easy-to-use, allowing for the implementation of a wide range of topology optimisation problems with a syntax that is near one-to-one with the mathematical notation. Furthermore, we implement automatic differentiation to help mitigate the bottleneck associated with the analytic derivation of sensitivities for complex problems. GridapTopOpt is capable of solving a range of benchmark and research topology optimisation problems with large numbers of degrees of freedom. This educational article demonstrates the usability and versatility of the package by describing the formulation and step-by-step implementation of several distinct topology optimisation problems. The driver scripts for these problems are provided and the package source code is available at https://github$.$com/zjwegert/GridapTopOpt.jl.

cs.MS↗