About

Andrea Bonito is a professor in the Department of Mathematics at Texas A&M University. Her research interests focus on geometric partial differential equations (PDEs) and approximation methods in high dimensions. She leads a research group that works on the numerical approximation of solutions to PDEs, including fractional differential operators and complex fluid dynamics. Her group includes PhD students working on topics such as the preasymptotic modeling of thin structures with folding, numerical approximation of time-dependent fractional diffusion with drift, and numerical simulations related to surface quasi-geostrophic dynamics and electroconvection. Andrea Bonito's research also involves the development of custom software tools and applications to support these computational studies. She actively mentors graduate and undergraduate students interested in theoretical and computational aspects of PDEs, particularly in finite element approximation methods. Her group welcomes collaboration and participation from students and postdoctoral fellows interested in advancing the numerical analysis and simulation of PDEs with incomplete data or complex boundary conditions.

Research topics

• Mathematical analysis
• Physics
• Geometry
• Mathematics

Selected publications

• Alleviating missing boundary conditions in elliptic partial differential equations using interior point measurements

  arXiv (Cornell University) · 2025-11-25

  preprintOpen access1st authorCorresponding

  We consider an optimal recovery problem for the Poisson problem when the boundary data is unknown. Compensating information is provided in the form of a finite number of measurements of the solution. A finite element algorithm for this problem was given in Binev et al. (2024), where measurements were assumed to be either bounded linear functionals of the solution or point measurements at locations lying anywhere in the closure of the computational domain. In contrast, we focus on the case of point measurements at locations lying in the interior of the domain. This lowers the regularity requirements placed on the solution. Also, a key ingredient in the recovery process is the finite element approximation of Riesz representers associated with the measurements. Our main result is a pointwise error estimate for the Riesz representers. We apply this to obtain improved estimates which measure the performance of the recovery algorithm in various norms.

  PublisherOA PDFDOI

• Babuška's paradox in a nonlinear bending-folding model

  ArXiv.org · 2025-03-21

  preprintOpen access

  The Babuška or plate paradox concerns the failure of convergence when a domain with curved boundary is approximated by polygonal domains in linear bending problems with simply supported boundary conditions. It can be explained via a boundary integral representation of the total Gaussian curvature that is part of the Kirchhoff--Love bending energy. It is shown that the paradox also occurs for a nonlinear bending-folding model which enforces vanishing Gaussian curvature. A simple remedy that is compatible with simplicial finite element methods to avoid incorrect convergence is devised.

  PublisherOA PDFDOI

• Optimal learning

  CALCOLO · 2024-02-19 · 5 citations

  article
  PublisherDOI

• Approximating Partial Differential Equations without Boundary Conditions

  2024-01-01

  book-chapter1st authorCorresponding
  PublisherDOI

• Adaptive Finite Element Methods

  arXiv (Cornell University) · 2024-02-11

  preprintOpen access1st authorCorresponding

  This is a survey on the theory of adaptive finite element methods (AFEMs), which are fundamental in modern computational science and engineering but whose mathematical assessment is a formidable challenge. We present a self-contained and up-to-date discussion of AFEMs for linear second order elliptic PDEs and dimension d>1, with emphasis on foundational issues. After a brief review of functional analysis and basic finite element theory, including piecewise polynomial approximation in graded meshes, we present the core material for coercive problems. We start with a novel a posteriori error analysis applicable to rough data, which delivers estimators fully equivalent to the solution error. They are used in the design and study of three AFEMs depending on the structure of data. We prove linear convergence of these algorithms and rate-optimality provided the solution and data belong to suitable approximation classes. We also address the relation between approximation and regularity classes. We finally extend this theory to discontinuous Galerkin methods as prototypes of non-conforming AFEMs and beyond coercive problems to inf-sup stable AFEMs.

  PublisherOA PDFDOI

• Adaptive finite element methods

  Acta Numerica · 2024-07-01 · 38 citations

  articleOpen access1st authorCorresponding

  This is a survey of the theory of adaptive finite element methods (AFEMs), which are fundamental to modern computational science and engineering but whose mathematical assessment is a formidable challenge. We present a self-contained and up-to-date discussion of AFEMs for linear second-order elliptic PDEs and dimension d > 1 , with emphasis on foundational issues. After a brief review of functional analysis and basic finite element theory, including piecewise polynomial approximation in graded meshes, we present the core material for coercive problems. We start with a novel a posteriori error analysis applicable to rough data, which delivers estimators fully equivalent to the solution error. They are used in the design and study of three AFEMs depending on the structure of data. We prove linear convergence of these algorithms and rate-optimality provided the solution and data belong to suitable approximation classes. We also address the relation between approximation and regularity classes. We finally extend this theory to discontinuous Galerkin methods as prototypes of non-conforming AFEMs, and beyond coercive problems to inf-sup stable AFEMs.

  PublisherDOI

• Approximating partial differential equations without boundary conditions

  arXiv (Cornell University) · 2024-06-05

  preprintOpen access1st authorCorresponding

  We consider the problem of numerically approximating the solutions to an elliptic partial differential equation (PDE) for which the boundary conditions are lacking. To alleviate this missing information, we assume to be given measurement functionals of the solution. In this context, a near optimal recovery algorithm based on the approximation of the Riesz representers of these functionals in some intermediate Hilbert spaces is proposed and analyzed in [Binev et al. 2024]. Inherent to this algorithm is the computation of $H^s$, $s>1/2$, inner products on the boundary of the computational domain. We take advantage of techniques borrowed from the analysis of fractional diffusion problems to design and analyze a fully practical near optimal algorithm not relying on the challenging computation of $H^s$ inner products.

  PublisherOA PDFDOI

• Finite element methods for the stretching and bending of thin structures with folding

  Numerische Mathematik · 2024-10-29 · 1 citations

  article1st author
  PublisherDOI

• Gamma-convergent LDG method for large bending deformations of bilayer plates

  IMA Journal of Numerical Analysis · 2024-01-17 · 3 citations

  article1st authorCorresponding

  Abstract Bilayer plates are slender structures made of two thin layers of different materials. They react to environmental stimuli and undergo large bending deformations with relatively small actuation. The reduced model is a constrained minimization problem for the second fundamental form, with a given spontaneous curvature that encodes material properties, subject to an isometry constraint. We design a local discontinuous Galerkin (LDG) method, which imposes a relaxed discrete isometry constraint and controls deformation gradients at barycenters of elements. We prove $\varGamma $-convergence of LDG, design a fully practical gradient flow, which gives rise to a linear scheme at every step, and show energy stability and control of the isometry defect. We extend the $\varGamma $-convergence analysis to piecewise quadratic creases. We also illustrate the performance of the LDG method with several insightful simulations of large deformations, one including a curved crease.

  PublisherDOI

• Solving PDEs with Incomplete Information

  SIAM Journal on Numerical Analysis · 2024-05-29 · 3 citations

  article
  PublisherDOI

Recent grants

• CAREER: Explicit Adaptive Methods for Coupled Problems

  NSF · $405k · 2013–2018

• Finite Element Approximations of Developable Surfaces with Curved Folds

  NSF · $300k · 2021–2024

• Space and Time Adaptivity for Moving and Free Boundary Problems

  NSF · $137k · 2009–2013

• Finite Element Approximations of Bending Actuated Devices

  NSF · $272k · 2018–2022

Frequent coauthors

• Ricardo H. Nochetto

  University of Maryland, College Park

  34 shared

• Diane Guignard

  University of Ottawa

  17 shared

• Ronald DeVore

  Texas A&M University

  14 shared

• Wenyu Lei

  Huazhong University of Science and Technology

  13 shared

• Vivette Girault

  Laboratoire Jacques-Louis Lions

  12 shared

• Joseph E. Pasciak

  Texas A&M University

  12 shared

• Jean‐Luc Guermond

  11 shared

• Albert Cohen

  7 shared

Labs

• Andrea Bonito's LabPI

  Research in Complex Fluids, Geometric PDEs, and Fractional Operators

Education

• PhD, Mathematics

  École Polytechnique Fédérale de Lausanne

  2006

• BS and MS, Mathematics

  EPFL

  2002