42 citations · 132 across the 19 of their papers we have counts for
20 papers
Inexact Uzawa-Double Deep Ritz Method for Weak Adversarial Neural Networks
Emin Benny-Chacko, Ignacio Brevis, Luis Espath +1
Residual minimization in dual norms is central to Weak Adversarial Neural Network (WAN) approaches for solving partial differential equations (PDEs). This framework naturally leads…
Optimal Control of Thin-Film Flow on a Flexible Topography
S. Alrashidy, A. Kalogirou, D. Kalise +1
This work presents a mathematical model for the optimal control of thin-film flows over a flexible substrate influenced by an external force. The objective is to find the optimal d…
Neural network methods for Neumann series problems of Perron-Frobenius operators
T. Udomworarat, I. Brevis, M. Richter +2
Problems related to Perron-Frobenius operators (or transfer operators) have been extensively studied and applied across various fields. In this work, we propose neural network meth…
A bulk-surface continuum theory for fluid flows and phase segregation with finite surface thickness
Anne Boschman, Luis Espath, Kris van der Zee
In this continuum theory, we propose a mathematical framework to study the mechanical interplay of bulk-surfaces materials undergoing deformation and phase segregation. To this end…
A Shape-Newton Method for Free-boundary Problems Subject to The Bernoulli Boundary Condition
Yiyun Fan, John Billingham, Kristoffer van der Zee
We develop a shape-Newton method for solving generic free-boundary problems where one of the free-boundary conditions is governed by the Bernoulli equation. The Newton-like scheme…
Learning quantities of interest from parametric PDEs: An efficient neural-weighted Minimal Residual approach
Ignacio Brevis, Ignacio Muga, David Pardo +2
The efficient approximation of parametric PDEs is of tremendous importance in science and engineering. In this paper, we show how one can train Galerkin discretizations to efficien…
Thermodynamically consistent diffuse-interface mixture models of incompressible multicomponent fluids
M. ten Eikelder, K. van der Zee, D. Schillinger
In this paper we derive a class of thermodynamically consistent diffuse-interface mixture models of incompressible multicomponent fluids. The class of mixture models is fully compa…
Neural Control of Discrete Weak Formulations: Galerkin, Least-Squares and Minimal-Residual Methods with Quasi-Optimal Weights
Ignacio Brevis, Ignacio Muga, Kristoffer G. van der Zee
There is tremendous potential in using neural networks to optimize numerical methods. In this paper, we introduce and analyse a framework for the neural optimization of discrete we…
Linearisation of the Travel Time Functional in Porous Media Flows
Paul Houston, Connor J. Rourke, Kristoffer G. Van der Zee
The travel time functional measures the time taken for a particle trajectory to travel from a given initial position to the boundary of the domain. Such evaluation is paramount in…
A unified framework for Navier-Stokes Cahn-Hilliard models with non-matching densities
M. ten Eikelder, K. van der Zee, I. Akkerman +1
Over the last decades, many diffuse-interface Navier-Stokes Cahn-Hilliard models with non-matching densities have appeared in the literature. These models claim to describe the sam…
Data-Driven Finite Elements Methods: Machine Learning Acceleration of Goal-Oriented Computations
Ignacio Brevis, Ignacio Muga, Kristoffer G. van der Zee
We introduce the concept of data-driven finite element methods. These are finite-element discretizations of partial differential equations (PDEs) that resolve quantities of interes…
Gibbs Phenomena for -Best Approximation in Finite Element Spaces -- Some Examples
Paul Houston, Sarah Roggendorf, Kristoffer G. van der Zee
Recent developments in the context of minimum residual finite element methods are paving the way for designing finite element methods in non-standard function spaces. This, in part…
Eliminating Gibbs Phenomena: A Non-linear Petrov-Galerkin Method for the Convection-Diffusion-Reaction Equation
Paul Houston, Sarah Roggendorf, Kristoffer G. van der Zee
In this article we consider the numerical approximation of the convection-diffusion-reaction equation. One of the main challenges of designing a numerical method for this problem i…
The Discrete-Dual Minimal-Residual Method (DDMRes) for Weak Advection-Reaction Problems in Banach spaces
Ignacio Muga, Matthew J. W. Tyler, Kristoffer G. van der Zee
We propose and analyse a minimal-residual method in discrete dual norms for approximating the solution of the advection-reaction equation in a weak Banach-space setting. The weak f…
A-posteriori error estimation and adaptivity for nonlinear parabolic equations using IMEX-Galerkin discretization of primal and dual equations
Xunxun Wu, Kristoffer van der Zee, Gorkem Simsek +1
While many methods exist to discretize nonlinear time-dependent partial differential equations (PDEs), the rigorous estimation and adaptive control of their discretization errors r…
Massively parallel-in-space-time, adaptive finite element framework for non-linear parabolic equations
Robert Dyja, Baskar Ganapathysubramanian, Kristoffer G. van der Zee
We present an adaptive methodology for the solution of (linear and) non-linear time dependent problems that is especially tailored for massively parallel computations. The basic co…
Diffuse-Interface Two-Phase Flow Models with Different Densities: A New Quasi-Incompressible Form and a Linear Energy-Stable Method
Mahnaz Shokrpour Roudbari, Gorkem Simsek, E. Harald van Brummelen +1
While various phase-field models have recently appeared for two-phase fluids with different densities, only some are known to be thermodynamically consistent, and practical stable…
Discretization of Linear Problems in Banach Spaces: Residual Minimization, Nonlinear Petrov-Galerkin, and Monotone Mixed Methods
Ignacio Muga, Kristoffer G. van der Zee
This work presents a comprehensive discretization theory for abstract linear operator equations in Banach spaces. The fundamental starting point of the theory is the idea of residu…
An abstract analysis of optimal goal-oriented adaptivity
Michael Feischl, Dirk Praetorius, Kristoffer G. van der Zee
We provide an abstract framework for optimal goal-oriented adaptivity for finite element methods and boundary element methods in the spirit of [Carstensen et al., Comput. Math. App…
Discontinuities without discontinuity: The Weakly-enforced Slip Method
G. J. van Zwieten, E. H. van Brummelen, K. G. van der Zee +2
Tectonic faults are commonly modelled as Volterra or Somigliana dislocations in an elastic medium. Various solution methods exist for this problem. However, the methods used in pra…