arXiv subjects

Santiago Badia

Publications and source records attributed to Santiago Badia.

At least 19 recordsLinked to original sources

arXiv2026

Topology optimisation of adsorption beds for cost-effective direct air capture

The performance of adsorption systems is sensitive to the geometry of the adsorbent. To achieve feasibility for large-scale implementation of CO2 direct air capture systems in line with the latest Intergovernmental Panel on Climate Change objectives, significant improvements to current adsorption systems must be made. In this study, we employ a topology optimization to the macrostructure of a swing adsorption bed using an experimentally verified numerical model. We unveil a simple blunted cone design, which achieves an estimated cost range of \$49-116/t-CO2 removed from the atmosphere for the system. This represents an improvement over the benchmark monolith design, which incurs a cost of \$75-140/t-CO2 considering the same process parameters.

cs.CE↗

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

GridapGeosciences.jl: A Julia finite element package for partial differential equations on general manifolds

We present GridapGeosciences$.$jl, a new parallel distributed-memory Julia package for the numerical approximation of partial differential equations on general manifolds. Our abstract approach to generating discrete manifold domains relies on mesh refinement in general, and is implemented within the Gridap finite element library, whose high-level interface enables seamless definition of both intrinsic and extrinsic variational formulations. GridapGeosciences$.$jl features the cubed sphere manifold, which is used in this study to showcase the functionality of GridapGeosciences$.$jl in different applications. The first application involves scalar transport where we develop intrinsic formulations of common stabilisation terms, which are shown to be equivalent to well-known extrinsic stabilisation formalisations. Such stabilisation terms are used to develop an intrinsic formulation of the two-dimensional thermal shallow water equations that includes upwinded numerical fluxes. The final application is a linearisation of the Boussinesq equations where we demonstrate the capacity of GridapGeosciences$.$jl to capture both surface and vertical dynamics on a three-dimensional manifold.

math.NA↗

arXiv2026

An intrinsic finite element framework for scalar- and vector-valued partial differential equations on general manifolds

We present an intrinsic finite element framework for the numerical approximation of scalar- and vector-valued partial differential equations on general manifolds described by atlases of charts. Weak formulations and their discretisation are expressed exclusively in the flat parametric space of the manifold, where the exact geometry enters only through the metric tensor that is evaluated at quadrature points. Conforming finite element spaces of arbitrary order for the whole de Rham complex are obtained on multi-chart atlases through purely topological inter-chart gluing. We develop a nodal change of basis construction for vector-valued spaces on multi-chart atlases, where nodal degrees of freedom couple charts with incompatible coordinate systems through transmission maps. Inter-chart continuity holds at the nodes with jumps of the order of the approximation error, and the resulting discrete field is exactly tangent to the manifold. Optimal convergence rates for this method are observed numerically for the surface Stokes problem; no a priori error analysis is provided. Numerical experiments on the cubed sphere manifold in two and three dimensions for the Hodge Laplacian problems demonstrate the intrinsic framework is free of the geometric consistency errors for a sufficient quadrature degree, and that intrinsic assembly is substantially cheaper than its extrinsic counterpart. Further considering the rotating shallow water equations that include orography as a perturbation of the geometry demonstrates that mass is conserved exactly, and energy is conserved up to the time discretisation error.

math.NA↗

arXiv2026

Divergence-free unfitted finite element discretisations for the Darcy problem

We develop an unfitted compatible finite element discretisation for the Darcy problem based on $H(\mathrm{div})$-conforming flux spaces and discontinuous pressure spaces. The method is designed to preserve pointwise discrete mass conservation while remaining robust in the presence of arbitrarily small cut cells arising from unfitted meshes. Robustness is achieved by combining an $L^2$-stabilisation of the flux with an additional mixed-term stabilisation that enhances pressure control without destroying the local conservation structure. We consider both cell-wise (bulk) and face-based ghost-penalty realisations of the stabilisation. Mixed boundary conditions are handled by weak imposition of both flux and pressure traces on unfitted boundaries. We prove stability and a priori error estimates with constants independent of the cut configuration, and establish pressure-robust flux error bounds in the case of pure pressure boundary conditions. We also introduce an augmented Lagrangian variant that improves control of the conservation constraint and is amenable to efficient preconditioning strategies. Numerical experiments for a range of cut configurations, boundary-condition regimes and parameter choices confirm the theoretical results, demonstrating optimal convergence, cut-independent conditioning and mass conservation up to solver tolerance.

math.NA↗

arXiv2026

Adaptive anisotropic composite quadratures for residual minimisation in neural PDE approximations

We study the role of numerical quadrature in residual-minimisation methods for neural network approximation of partial differential equations. We first present an abstract error framework that separates approximation, quadrature and optimisation errors, and derive a nonlinear Strang-type estimate quantifying how inaccuracies in the discrete loss affect the final approximation. Motivated by this analysis, we propose an anisotropic adaptive composite quadrature strategy that controls the relative quadrature error of the residual loss using richer reference quadratures and bisection-based refinement. We then introduce a refresh-based training methodology that rebuilds the quadrature only when an online error indicator exceeds a prescribed threshold, balancing accuracy and computational cost. Numerical experiments on a range of benchmark problems show that the proposed approach narrows the gap between training and reference losses, uses quadrature points more efficiently and delivers strong approximation accuracy relative to non-adaptive quadrature strategies.

math.NA↗

arXiv2026

Four-field mixed finite elements for incompressible nonlinear elasticity

We present a stable finite element method for incompressible nonlinear elasticity based on a four-field mixed formulation involving the displacement, displacement gradient, first Piola--Kirchhoff stress and pressure. Unlike existing four-field mixed formulations, such as the compatible strain mixed finite element method (CSFEM), the proposed approach employs a discontinuous displacement field and requires no stabilisation in either 2D or 3D. A Newton--Raphson linearisation is derived and finite element pairs satisfying the relevant inf-sup conditions are identified. To recover accurate continuous displacement fields, an efficient postprocessing technique is further introduced. We establish the well-posedness of the linearised continuous problem together with a priori error estimates for the discrete formulation. Extensive numerical experiments in both 2D and 3D demonstrate optimal or even super convergence rates and enhanced robustness, particularly in 3D where CSFEM typically requires stabilisation.

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

Reduced basis solvers for unfitted methods on parameterized domains

In this paper, we present a unified framework for reduced basis approximations of parametrized partial differential equations defined on parameter-dependent domains. Our approach combines unfitted finite element methods with both classical and tensor-based reduced basis techniques -- particularly the tensor-train reduced basis method -- to enable efficient and accurate model reduction on general geometries. To address the challenge of reconciling geometric variability with fixed-dimensional snapshot representations, we adopt a deformation-based strategy that maps a reference configuration to each parameterized domain. Furthermore, we introduce a localization procedure to construct dictionaries of reduced subspaces and hyper-reduction approximations, which are obtained via matrix discrete empirical interpolation in our work. We extend the proposed framework to saddle-point problems by adapting the supremizer enrichment strategy to unfitted methods and deformed configurations, demonstrating that the supremizer operator can be defined on the reference configuration without loss of stability. Numerical experiments on two- and three-dimensional problems -- including Poisson, linear elasticity, incompressible Stokes and Navier-Stokes equations -- demonstrate the flexibility, accuracy and efficiency of the proposed methodology.

math.NA↗

arXiv2025

GridapROMs.jl: Efficient reduced order modelling in the Julia programming language

In this paper, we introduce GridapROMs, a Julia-based library for the numerical approximation of parameterized partial differential equations (PDEs) using a comprehensive suite of linear reduced order models (ROMs). The library is designed to be extendable and productive, leveraging an expressive high-level API built on the Gridap PDE solver backend, while achieving high performance through Julia's just-in-time compiler and advanced lazy evaluation techniques. GridapROMs is PDE-agnostic, enabling its application to a wide range of problems, including linear, nonlinear, single-field, multi-field, steady, and unsteady equations. This work details the library's key innovations, implementation principles, and core components, providing usage examples and demonstrating its capabilities by solving a fluid dynamics problem modeled by the Navier-Stokes equations in a 3D geometry.

math.NA↗

arXiv2025

A convergence framework for energy minimisation of linear self-adjoint elliptic PDEs in nonlinear approximation spaces

Recent years have seen the emergence of nonlinear methods for solving partial differential equations (PDEs), such as physics-informed neural networks (PINNs). While these approaches often perform well in practice, their theoretical analysis remains limited, especially regarding convergence guarantees. This work develops a general optimisation framework for energy minimisation problems arising from linear self-adjoint elliptic PDEs, formulated over nonlinear but analytically tractable approximation spaces. The framework accommodates a natural split between linear and nonlinear parameters and supports hybrid optimisation strategies: linear variables are updated via linear solves or steepest descent, while nonlinear variables are handled using constrained projected descent. We establish both local and global convergence of the resulting algorithm under modular structural assumptions on the discrete energy functional, including differentiability, boundedness, regularity, and directional convexity. These assumptions are stated in an abstract form, allowing the framework to apply to a broad class of nonlinear approximation manifolds. In a companion paper [Magueresse, Badia (2025, arXiv:2508.17705)], we introduce a concrete instance of such a space based on overlapping free-knot tensor-product B-splines, which satisfies the required assumptions and enables geometrically adaptive solvers with rigorous convergence guarantees.

math.NA↗

arXiv2025

Energy minimisation using overlapping tensor-product free-knot B-splines

Accurately solving PDEs with localised features requires refined meshes that adapt to the solution. Traditional numerical methods, such as finite elements, are linear in nature and often ineffective for such problems, as the mesh is not tailored to the solution. Adaptive strategies, such as $h$- and $p$-refinement, improve efficiency by sequentially refining the mesh based on a posteriori error estimates. However, these methods are geometrically rigid -- limited to specific refinement rules -- and require solving the problem on a sequence of adaptive meshes, which can be computationally expensive. Moreover, the design of effective a posteriori error estimates is problem-dependent and non-trivial. In this work, we study a specific nonlinear approximation scheme based on overlapping tensor-product free-knot B-spline patches, where knot positions act as nonlinear parameters controlling the geometry of the discretisation. We analyse the corresponding energy minimisation problem for linear, self-adjoint elliptic PDEs, showing that, under a mild mesh size condition, the discrete energy satisfies the structural properties required for the local and global convergence of the constrained optimisation scheme developed in our companion work [Magueresse, Badia (2025, arXiv:2508.17687)]. This establishes a direct connection between the two analyses: the adaptive free-knot B-spline space considered here fits into the abstract framework, ensuring convergence of projected gradient descent for the joint optimisation of knot positions and coefficients. Numerical experiments illustrate the method's efficiency and its ability to capture localised features with significantly fewer degrees of freedom than standard finite element discretisations.

math.NA↗

arXiv2025

A velocity-vorticity-pressure formulation for the steady Navier--Stokes--Brinkman--Forchheimer problem

The flow of incompressible fluid in highly permeable porous media in vorticity - velocity - Bernoulli pressure form leads to a double saddle-point problem in the Navier--Stokes--Brinkman--Forchheimer equations. The paper establishes, for small sources, the existence of solutions on the continuous and discrete level of lowest-order piecewise divergence-free Crouzeix--Raviart finite elements. The vorticity employs a vector version of the pressure space with normal and tangential velocity jump penalisation terms. A simple Raviart--Thomas interpolant leads to pressure-robust a priori error estimates. An explicit residual-based a posteriori error estimate allows for efficient and reliable a posteriori error control. The efficiency for the Forchheimer nonlinearity requires a novel discrete inequality of independent interest. The implementation is based upon a light-weight forest-of-trees data structure handled by a highly parallel set of adaptive mesh refining algorithms. Numerical simulations reveal robustness of the a posteriori error estimates and improved convergence rates by adaptive mesh-refining.

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↗

arXiv2025

Unfitted finite element interpolated neural networks

We present a novel approach that integrates unfitted finite element methods and neural networks to approximate partial differential equations on complex geometries. Easy-to-generate background meshes (e.g., a simple Cartesian mesh) that cut the domain boundary (i.e., they do not conform to it) are used to build suitable trial and test finite element spaces. The method seeks a neural network that, when interpolated onto the trial space, minimises a discrete norm of the weak residual functional on the test space associated to the equation. As with unfitted finite elements, essential boundary conditions are weakly imposed by Nitsche's method. The method is robust to variations in Nitsche coefficient values, and to small cut cells. We experimentally demonstrate the method's effectiveness in solving both forward and inverse problems across various 2D and 3D complex geometries, including those defined by implicit level-set functions and explicit stereolithography meshes. For forward problems with smooth analytical solutions, the trained neural networks achieve several orders of magnitude smaller $H^1$ errors compared to their interpolation counterparts. These interpolations also maintain expected $h$- and $p$-convergence rates. Using the same amount of training points, the method is faster than standard PINNs (on both GPU and CPU architectures) while achieving similar or superior accuracy. Moreover, using a discrete dual norm of the residual (achieved by cut cell stabilisation) remarkably accelerates neural network training and further enhances robustness to the choice of Nitsche coefficient values. The experiments also show the method's high accuracy and reliability in solving inverse problems, even with incomplete observations.

math.NA↗

arXiv2025

A discrete trace theory for non-conforming polytopal hybrid discretisation methods

In this work we develop a discrete trace theory that spans non-conforming hybrid discretization methods and holds on polytopal meshes. A notion of a discrete trace seminorm is defined, and trace and lifting results with respect to a discrete $H^1$-seminorm on the hybrid fully discrete space are proven. Building on these results we also prove a truncation estimate for piecewise polynomials in the discrete trace seminorm. Finally, we conduct two numerical tests in which we compute the proposed discrete operators and investigate their spectrum to verify the theoretical analysis. The development of this theory is motivated by the design and analysis of preconditioners for hybrid methods, e.g., of substructuring domain decomposition type.

math.NA↗

arXiv2025

A tensor-train reduced basis solver for parameterized partial differential equations on Cartesian grids

In this manuscript, we introduce the tensor-train reduced basis method, a novel projection-based reduced-order model designed for the efficient solution of parameterized partial differential equations. While reduced-order models are widely used for their computational efficiency compared to full-order models, they often involve significant offline computational costs. Our proposed approach mitigates this limitation by leveraging the tensor train format to efficiently represent high-dimensional finite element quantities. This method offers several advantages, including a reduced number of operations for constructing the reduced subspaces, a cost-effective hyper-reduction strategy for assembling the PDE residual and Jacobian, and a lower dimensionality of the projection subspaces for a given accuracy. We provide a posteriori error estimates to validate the accuracy of the method and evaluate its computational performance on benchmark problems, including the Poisson equation, heat equation, and transient linear elasticity in two- and three-dimensional domains. Although the current framework is restricted to problems defined on Cartesian grids, we anticipate that it can be extended to arbitrary shapes by integrating the tensor-train reduced basis method with unfitted finite element techniques.

math.NA↗