About

Alan Demlow is a professor at Texas A&M University in the College of Arts and Sciences, within the Department of Mathematics. His research interests focus on the numerical analysis of partial differential equations. He is associated with the Institute for Applied Mathematics and Computational Science and is involved in various programs and outreach activities related to mathematics education and research. His office is located in Blocker 507D, and he can be contacted via email at demlow@tamu.edu.

Research topics

• Mathematical analysis
• Geometry
• Mathematics
• Physics
• Applied mathematics
• Mathematical optimization
• Mechanics

Selected publications

• A Taylor–Hood Finite Element Method for the Surface Stokes Problem Without Penalization

  SIAM Journal on Numerical Analysis · 2026-04-30

  articleOpen access1st authorCorresponding

  Finite element approximation of the velocity-pressure formulation of the surfaces Stokes equations is challenging because it is typically not possible to enforce both tangentiality and $H^1$ conformity of the velocity field. Most previous works concerning finite element methods (FEMs) for these equations thus have weakly enforced one of these two constraints by penalization or a Lagrange multiplier formulation. Recently in [A tangential and penalty-free finite element method for the surface Stokes problem, SINUM 62(1):248-272, 2024], the authors constructed a surface Stokes FEM based on the MINI element which is tangentiality conforming and $H^1$ nonconforming, but possesses sufficient weak continuity properties to circumvent the need for penalization. The key to this method is construction of velocity degrees of freedom lying on element edges and vertices using an auxiliary Piola transform. In this work we extend this methodology to construct Taylor-Hood surface FEMs. The resulting method is shown to achieve optimal-order convergence when the edge degrees of freedom for the velocity space are placed at Gauss-Lobatto nodes. Numerical experiments confirm that this nonstandard placement of nodes is necessary to achieve optimal convergence orders.

  PublisherOA PDFDOI

• A mixed quasi-trace surface finite element method for the Laplace–Beltrami problem

  Journal of Numerical Mathematics · 2026-03-10

  articleOpen access1st authorCorresponding

  Abstract Trace finite element methods have become a popular option for solving surface partial differential equations, especially in problems where surface and bulk effects are coupled. In such methods a surface mesh is formed by approximately intersecting the continuous surface on which the PDE is posed with a three-dimensional (bulk) tetrahedral mesh. In H 1 -conforming trace methods, the surface finite element space is obtained by restricting a bulk finite element space to the surface mesh. It is not clear how to carry out a similar procedure in order to obtain other important types of finite element spaces such as H (div)-conforming spaces. Following previous work of Olshanskii, Reusken, and Xu on H 1 -conforming methods, we develop a “quasi-trace” mixed method for the Laplace–Beltrami problem. The finite element mesh is taken to be the intersection of the surface with a regular tetrahedral bulk mesh as previously described, resulting in a surface triangulation that is highly unstructured and anisotropic but satisfies a classical maximum angle condition. The mixed method is then employed on this mesh. Optimal error estimates with respect to the bulk mesh size are proved along with superconvergent estimates for the projection of the scalar error and a postprocessed scalar approximation. Higher-order surface approximations are also considered.

  PublisherDOI

• Schatz, Alfred (Al) H.

  eCommons (Cornell University) · 2025-10-16

  other

  Memorial Statement for Alfred (Al) H. Schatz who died in 2024. The memorial statements contained herein were prepared by the Office of the Dean of the University Faculty of Cornell University to honor its faculty for their service to the university.

  Publisher

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

  arXiv (Cornell University) · 2025-11-25

  preprintOpen access

  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

• A Tangential and Penalty-Free Finite Element Method for the Surface Stokes Problem

  SIAM Journal on Numerical Analysis · 2024 · 8 citations

  1st authorCorresponding
  • Mathematics
  • Mathematical analysis
  • Applied mathematics

  .Surface Stokes and Navier–Stokes equations are used to model fluid flow on surfaces. They have attracted significant recent attention in the numerical analysis literature because approximation of their solutions poses significant challenges not encountered in the Euclidean context. One challenge comes from the need to simultaneously enforce tangentiality and \(H^1\) conformity (continuity) of discrete vector fields used to approximate solutions in the velocity-pressure formulation. Existing methods in the literature all enforce one of these two constraints weakly either by penalization or by use of Lagrange multipliers. Missing so far is a robust and systematic construction of surface Stokes finite element spaces which employ nodal degrees of freedom, including MINI, Taylor–Hood, Scott–Vogelius, and other composite elements which can lead to divergence-conforming or pressure-robust discretizations. In this paper we construct surface MINI spaces whose velocity fields are tangential. They are not \(H^1\)-conforming, but do lie in \(H(\textrm{div})\) and do not require penalization to achieve optimal convergence rates. We prove stability and optimal-order energy-norm convergence of the method and demonstrate optimal-order convergence of the velocity field in \(L_2\) via numerical experiments. The core advance in the paper is the construction of nodal degrees of freedom for the velocity field. This technique also may be used to construct surface counterparts to many other standard Euclidean Stokes spaces, and we accordingly present numerical experiments indicating optimal-order convergence of nonconforming tangential surface Taylor–Hood \(\mathbb{P}^2-\mathbb{P}^1\) elements.Keywordssurface Stokes equationfinite element methodMINI elementMSC codes65N1265N1565N30

  PublisherDOI

• A mixed quasi-trace surface finite element method for the Laplace-Beltrami problem

  arXiv (Cornell University) · 2023-09-21

  preprintOpen access1st authorCorresponding

  Trace finite element methods have become a popular option for solving surface partial differential equations, especially in problems where surface and bulk effects are coupled. In such methods a surface mesh is formed by approximately intersecting the continuous surface on which the PDE is posed with a three-dimensional (bulk) tetrahedral mesh. In classical $H^1$-conforming trace methods, the surface finite element space is obtained by restricting a bulk finite element space to the surface mesh. It is not clear how to carry out a similar procedure in order to obtain other important types of finite element spaces such as $H({\rm div})$-conforming spaces. Following previous work of Olshanskii, Reusken, and Xu on $H^1$-conforming methods, we develop a ``quasi-trace'' mixed method for the Laplace-Beltrami problem. The finite element mesh is taken to be the intersection of the surface with a regular tetrahedral bulk mesh as previously described, resulting in a surface triangulation that is highly unstructured and anisotropic but satisfies a classical maximum angle condition. The mixed method is then employed on this mesh. Optimal error estimates with respect to the bulk mesh size are proved along with superconvergent estimates for the projection of the scalar error and a postprocessed scalar approximation.

  PublisherOA PDFDOI

• A tangential and penalty-free finite element method for the surface Stokes problem

  arXiv (Cornell University) · 2023-07-04 · 1 citations

  preprintOpen access1st authorCorresponding

  Surface Stokes and Navier-Stokes equations are used to model fluid flow on surfaces. They have attracted significant recent attention in the numerical analysis literature because approximation of their solutions poses significant challenges not encountered in the Euclidean context. One challenge comes from the need to simultaneously enforce tangentiality and $H^1$ conformity (continuity) of discrete vector fields used to approximate solutions in the velocity-pressure formulation. Existing methods in the literature all enforce one of these two constraints weakly either by penalization or by use of Lagrange multipliers. Missing so far is a robust and systematic construction of surface Stokes finite element spaces which employ nodal degrees of freedom, including MINI, Taylor-Hood, Scott-Vogelius, and other composite elements which can lead to divergence-conforming or pressure-robust discretizations. In this paper we construct surface MINI spaces whose velocity fields are tangential. They are not $H^1$-conforming, but do lie in $H({\rm div})$ and do not require penalization to achieve optimal convergence rates. We prove stability and optimal-order energy-norm convergence of the method and demonstrate optimal-order convergence of the velocity field in $L_2$ via numerical experiments. The core advance in the paper is the construction of nodal degrees of freedom for the velocity field. This technique also may be used to construct surface counterparts to many other standard Euclidean Stokes spaces, and we accordingly present numerical experiments indicating optimal-order convergence of nonconforming tangential surface Taylor-Hood $\mathbb{P}^2-\mathbb{P}^1$ elements.

  PublisherOA PDFDOI

• Maximum norm <i>a posteriori</i> error estimates for convection–diffusion problems

  IMA Journal of Numerical Analysis · 2023-02-20 · 4 citations

  articleOpen access1st author

  Abstract We prove residual-type a posteriori error estimates in the maximum norm for a linear scalar elliptic convection–diffusion problem that may be singularly perturbed. Similar error analysis in the energy norm by Verfürth indicates that a dual norm of the convective derivative of the error must be added to the natural energy norm in order for the natural residual estimator to be reliable and efficient. We show that the situation is similar for the maximum norm. In particular, we define a mesh-dependent weighted seminorm of the convective error, which functions as a maximum-norm counterpart to the dual norm used in the energy norm setting. The total error is then defined as the sum of this seminorm, the maximum norm of the error and data oscillation. The natural maximum norm residual error estimator is shown to be equivalent to this total error notion, with constant independent of singular perturbation parameters. These estimates are proved under the assumption that certain natural estimates hold for the Green’s function for the problem at hand. Numerical experiments confirm that our estimators effectively capture the maximum-norm error behavior for singularly perturbed problems, and can effectively drive adaptive refinement in order to capture layer phenomena.

  PublisherOA PDFDOI

• Maximum norm a posteriori error estimates for convection-diffusion problems

  arXiv (Cornell University) · 2022-07-17

  preprintOpen access1st authorCorresponding

  We prove residual-type a posteriori error estimates in the maximum norm for a linear scalar elliptic convection–diffusion problem that may be singularly perturbed. Similar error analysis in the energy norm by Verfürth indicates that a dual norm of the convective derivative of the error must be added to the natural energy norm in order for the natural residual estimator to be reliable and efficient. We show that the situation is similar for the maximum norm. In particular, we define a mesh-dependent weighted seminorm of the convective error, which functions as a maximum-norm counterpart to the dual norm used in the energy norm setting. The total error is then defined as the sum of this seminorm, the maximum norm of the error and data oscillation. The natural maximum norm residual error estimator is shown to be equivalent to this total error notion, with constant independent of singular perturbation parameters. These estimates are proved under the assumption that certain natural estimates hold for the Green’s function for the problem at hand. Numerical experiments confirm that our estimators effectively capture the maximum-norm error behavior for singularly perturbed problems, and can effectively drive adaptive refinement in order to capture layer phenomena.

  PublisherOA PDFDOI

• A Divergence-Conforming Finite Element Method for the Surface Stokes Equation

  SIAM Journal on Numerical Analysis · 2020-01-01 · 6 citations

  preprintOpen access

  The Stokes equation posed on surfaces is important in some physical models, but its numerical solution poses several challenges not encountered in the corresponding Euclidean setting. These include the fact that the velocity vector should be tangent to the given surface and the possible presence of degenerate modes (Killing fields) in the solution. We analyze a surface finite element method which provides solutions to these challenges. We consider an interior penalty method based on the well-known Brezzi-Douglas-Marini $H({\rm div})$-conforming finite element space. The resulting spaces are tangential to the surface, but require penalization of jumps across element interfaces in order to weakly maintain $H^1$ conformity of the velocity field. In addition our method exactly satisfies the incompressibility constraint in the surface Stokes problem. Secondly, we give a method which robustly filters Killing fields out of the solution. This problem is complicated by the fact that the dimension of the space of Killing fields may change with small perturbations of the surface. We first approximate the Killing fields via a Stokes eigenvalue problem and then give a method which is asymptotically guaranteed to correctly exclude them from the solution. The properties of our method are rigorously established via an error analysis and illustrated via numerical experiments.

  PublisherOA PDFDOI

Recent grants

• Problems in mathematical foundations of adaptive finite element methods

  NSF · $167k · 2014–2017

• PostDoctoral Research Fellowship

  NSF · $108k · 2003–2007

• Adaptive FEM for elliptic and parabolic problems

  NSF · $152k · 2010–2014

• Adaptive FEM for controlling pointwise errors and level sets

  NSF · $113k · 2007–2011

• Topics in Mathematical Theory of Adaptive Finite Element Methods

  NSF · $180k · 2017–2021

Frequent coauthors

• Andrea Bonito

  Texas A&M University

  11 shared

• Johnny Guzmán

  Brown University

  4 shared

• Andrea Bonito

  Texas A&M University

  3 shared

• Rob Stevenson

  3 shared

• Charalambos Makridakis

  3 shared

• Martin W. Licht

  3 shared

• A. H. Schatz

  3 shared

• Anil N. Hirani

  University of Illinois Urbana-Champaign

  3 shared