105 citations · 358 across the 18 of their papers we have counts for
19 papers
Uniform Preconditioners for High Order Finite Element Approximations of Planar Linear Elasticity
Mark Ainsworth, Charles Parker
A new preconditioner is developed for high order finite element approximation of linear elastic problems on triangular meshes in two dimensions. The new preconditioner results in a…
Statically Condensed Iterated Penalty Method for High Order Finite Element Discretizations of Incompressible Flow
Mark Ainsworth, Charles Parker
We introduce and analyze a Statically Condensed Iterated Penalty (SCIP) method for solving incompressible flow problems discretized with th-order Scott-Vogelius elements. While…
A Mass Conserving Mixed hp-FEM Scheme for Stokes Flow. Part III: Implementation and Preconditioning
Mark Ainsworth, Charles Parker
This is the third part in a series on a mass conserving, high order, mixed finite element method for Stokes flow. In this part, we study a block-diagonal preconditioner for the ind…
Galerkin Neural Networks: A Framework for Approximating Variational Equations with Error Control
Mark Ainsworth, Justin Dong
We present a new approach to using neural networks to approximate the solutions of variational equations, based on the adaptive construction of a sequence of finite-dimensional sub…
Plateau Phenomenon in Gradient Descent Training of ReLU networks: Explanation, Quantification and Avoidance
Mark Ainsworth, Yeonjong Shin
The ability of neural networks to provide `best in class' approximation across a wide range of applications is well-documented. Nevertheless, the powerful expressivity of neural ne…
A Simple Approach to Reliable and Robust A Posteriori Error Estimation for Singularly Perturbed Problems
Mark Ainsworth, Tomas Vejchodsky
A simple flux reconstruction for finite element solutions of reaction-diffusion problems is shown to yield fully computable upper bounds on the energy norm of error in an approxima…
Dispersive behavior of an energy-conserving discontinuous Galerkin method for the one-way wave equation
Mark Ainsworth, Guosheng Fu
The dispersive behavior of the recently proposed energy-conserving discontinuous Galerkin (DG) method by Fu and Shu [10] is analyzed and compared with the classical centered and up…
Bernstein-Bezier Bases for Tetrahedral Finite Elements
Mark Ainsworth, Guosheng Fu
We present a new set of basis functions for H(curl)-conforming, H(div)-conforming, and L2 -conforming finite elements of arbitrary order on a tetrahedron. The basis functions are e…
What Is the Fractional Laplacian?
Anna Lischke, Guofei Pang, Mamikon Gulian +8
The fractional Laplacian in R^d has multiple equivalent characterizations. Moreover, in bounded domains, boundary conditions must be incorporated in these characterizations in math…
Hybrid Finite Element - Spectral Method for the Fractional Laplacian: Approximation Theory and Efficient Solver
Mark Ainsworth, Christian Glusa
A numerical scheme is presented for approximating fractional order Poisson problems in two and three dimensions. The scheme is based on reformulating the original problem posed ove…
Aspects of an adaptive finite element method for the fractional Laplacian: a priori and a posteriori error estimates, efficient implementation and multigrid solver
Mark Ainsworth, Christian Glusa
We develop all of the components needed to construct an adaptive finite element code that can be used to approximate fractional partial differential equations, on non-trivial domai…
Towards an Efficient Finite Element Method for the Integral Fractional Laplacian on Polygonal Domains
Mark Ainsworth, Christian Glusa
We explore the connection between fractional order partial differential equations in two or more spatial dimensions with boundary integral operators to develop techniques that enab…
Fully computable a posteriori error bounds for hybridizable discontinuous Galerkin finite element approximations
Mark Ainsworth, Guosheng Fu
We derive a posteriori error estimates for the hybridizable discontinuous Galerkin (HDG) methods, including both the primal and mixed formulations, for the approximation of a linea…
A lowest-order composite finite element exact sequence on pyramids
Mark Ainsworth, Guosheng Fu
Composite basis functions for pyramidal elements on the spaces , , and are presented. In particular, we construct the lowes…
Is the Multigrid Method Fault Tolerant? The Multilevel Case
Mark Ainsworth, Christian Glusa
Computing at the exascale level is expected to be affected by a significantly higher rate of faults, due to increased component counts as well as power considerations. Therefore, c…
Is the Multigrid Method Fault Tolerant? The Two-Grid Case
Mark Ainsworth, Christian Glusa
The predicted reduced resiliency of next-generation high performance computers means that it will become necessary to take into account the effects of randomly occurring faults on…
Computing the Bézier Control Points of the Lagrangian Interpolant in Arbitrary Dimension
Mark Ainsworth, Manuel A. Sánchez
The Bernstein-Bézier form of a polynomial is widely used in the fields of computer aided geometric design, spline approximation theory and, more recently, for high order finite ele…
Discrete Extension Operators for Mixed finite element spaces on locally refined meshes
Mark Ainsworth, Johnny Guzmán, Francisco-Javier Sayas
The existence of uniformly bounded discrete extension operators is established for conforming Raviart-Thomas and Nédélec discretisations of and on locally refine…
Robust error bounds for finite element approximation of reaction-diffusion problems with non-constant reaction coefficient in arbitrary space dimension
Mark Ainsworth, Tomáš Vejchodský
We present a fully computable a posteriori error estimator for piecewise linear finite element approximations of reaction-diffusion problems with mixed boundary conditions and piec…