arXiv subjects

Marc Gerritsma

Publications and source records attributed to Marc Gerritsma.

At least 19 recordsLinked to original sources

arXiv2026

An Approximate Bounded Cochain Projection

This paper presents a construction of a projector from an infinite-dimensional Hilbert complex of differential $k$-forms onto a finite-dimensional piecewise polynomial sub-complex. We demonstrate that, on contractable domains, the proposed projector attains the three properties of the Bounded Cochain Projector, namely that the projector is idempotent, uniformly bounded in the Sobolev norm, and it commutes with the exterior derivative. On non-contractible domains, the projector remains idempotent and uniformly bounded, while the commuting property is satisfied up to arbitrary accuracy.

math.NA↗

arXiv2025

Space-time discretization for barotropic flow stemming from a multisymplectic variational formulation

This study proposes and analyses a novel higher-order, structure preserving discretization method for inviscid barotropic flows from a Lagrangian perspective. The method is built on a multisymplectic variational principle discretized over a full space-time domain. Flow variables are encoded on a staggered space-time mesh, leveraging the principles of mimetic spectral element discretization. Unlike standard Lagrangian methods, which are prone to mesh distortion, this framework computes fluid deformations in a fixed reference configuration and systematically maps them to the physical domain via the Piola-Kirchhoff stress. Further, the structure preserving design ensures that the discrete analogues of the fundamental conservation laws for mass, momentum, and energy are satisfied up to machine precision. The formulation also inherently handles low-Mach number flows without specialized preconditioning. Numerical experiments on expansion and compression flows confirm the accuracy, stability, and exact conservation properties of the discretization.

math.NA↗

arXiv2025

Optimal solutions employing an algebraic Variational Multiscale approach Part II: Application to Navier-Stokes

This work presents a non-linear extension of the high-order discretisation framework based on the Variational Multiscale (VMS) method previously introduced for steady linear problems. We build on the concept of an optimal projector defined via the symmetric part of the governing operator. Using this idea, we generalise the formulation to the two-dimensional incompressible Navier-Stokes equations. The approach maintains a clear separation between resolved and unresolved scales, with the fine-scale contribution approximated through the approximate Fine-Scale Greens' operator of the associated symmetric operator. This enables a consistent variational treatment of non-linearity while preserving high-order accuracy. We show that the method yields numerical solutions that closely approximate the optimal projection of the continuous/highly-resolved solution and inherits desirable conservation properties. Particularly, the formulation guarantees discrete conservation of mass, energy, and vorticity, where enstrophy conservation is also achieved when exact or over-integration is employed. Numerical results confirm the methodology's robustness and accuracy, while also demonstrating its computational cost advantage compared to the baseline Galerkin approach for the same accuracy.

math.NA↗

arXiv2024

Decoupled structure-preserving discretization of incompressible MHD equations with general boundary conditions

In the framework of a mixed finite element method, a structure-preserving formulation for incompressible magnetohydrodynamic (MHD) equations with general boundary conditions is proposed. A leapfrog-type temporal scheme fully decouples the fluid part from the Maxwell part by means of staggered discrete time sequences and, in doing so, partially linearizes the system. Conservation and dissipation properties of the formulation before and after the decoupling are analyzed. We demonstrate optimal spatial and second-order temporal accuracy, as well as conservation and dissipation properties, of the proposed method using manufactured solutions, and apply it to the benchmark Orszag-Tang and lid-driven cavity cases.

math.NA↗

arXiv2024

Optimal solutions employing an algebraic Variational Multiscale approach Part I: Steady Linear Problems

This work extends our previous study from S. Shrestha et al. (2024) by introducing a new abstract framework for Variational Multiscale (VMS) methods at the discrete level. We introduce the concept of what we define as the optimal projector and present a discretisation approach that yields a numerical solution closely approximating the optimal projection of the infinite-dimensional continuous solution. In this approach, the infinite-dimensional unresolved scales are approximated in a finite-dimensional subspace using the numerically computed Fine-Scale Greens' function of the underlying symmetric problem. The proposed approach involves solving the VMS problem on two separate meshes: a coarse mesh for the full PDE and a fine mesh for the symmetric part of the continuous differential operator. We consider the 1D and 2D steady advection-diffusion problems in both direct and mixed formulations as the test cases in this paper. We first present an error analysis of the proposed approach and show that the projected solution is achieved as the approximate Greens' function converges to the exact one. Subsequently, we demonstrate the working of this method where we show that it can exponentially converge to the chosen optimal projection. We note that the implementation of the present work employs the Mimetic Spectral Element Method (MSEM), although, it may be applied to other Finite/Spectral Element or Isogeometric frameworks. Furthermore, we propose that VMS should not be viewed as a stabilisation technique; instead, the base scheme should be inherently stable, with VMS enhancing the solution quality by supplementing the base scheme.

math.NA↗

arXiv2024

Predicting Open-Hole Laminates Failure Using Support Vector Machines With Classical and Quantum Kernels

Modeling open hole failure of composites is a complex task, consisting in a highly nonlinear response with interacting failure modes. Numerical modeling of this phenomenon has traditionally been based on the finite element method, but requires to tradeoff between high fidelity and computational cost. To mitigate this shortcoming, recent work has leveraged machine learning to predict the strength of open hole composite specimens. Here, we also propose using data-based models but to tackle open hole composite failure from a classification point of view. More specifically, we show how to train surrogate models to learn the ultimate failure envelope of an open hole composite plate under in-plane loading. To achieve this, we solve the classification problem via support vector machine (SVM) and test different classifiers by changing the SVM kernel function. The flexibility of kernel-based SVM also allows us to integrate the recently developed quantum kernels in our algorithm and compare them with the standard radial basis function (RBF) kernel. Finally, thanks to kernel-target alignment optimization, we tune the free parameters of all kernels to best separate safe and failure-inducing loading states. The results show classification accuracies higher than 90% for RBF, especially after alignment, followed closely by the quantum kernel classifiers.

cs.CE↗

arXiv2023

Construction and application of an algebraic dual basis and the Fine-Scale Greens' Function for computing projections and reconstructing unresolved scales

In this paper, we build on the work of [T. Hughes, G. Sangalli, VARIATIONAL MULTISCALE ANALYSIS: THE FINE-SCALE GREENS' FUNCTION, PROJECTION, OPTIMIZATION, LOCALIZATION, AND STABILIZED METHODS, SIAM Journal of Numerical Analysis, 45(2), 2007] dealing with the explicit computation of the Fine-Scale Green's function. The original approach chooses a set of functionals associated with a projector to compute the Fine-Scale Green's function. The construction of these functionals, however, does not generalise to arbitrary projections, higher dimensions, or Spectral Element methods. We propose to generalise the construction of the required functionals by using dual functions. These dual functions can be directly derived from the chosen projector and are explicitly computable. We show how to find the dual functions for both the $L^2$ and the $H^1_0$ projections. We then go on to demonstrate that the Fine-Scale Green's functions constructed with the dual basis functions consistently reproduce the unresolved scales removed by the projector. The methodology is tested using one-dimensional Poisson and advection-diffusion problems, as well as a two-dimensional Poisson problem. We present the computed components of the Fine-Scale Green's function, and the Fine-Scale Green's function itself. These results show that the method works for arbitrary projections, in arbitrary dimensions. Moreover, the methodology can be applied to any Finite/Spectral Element or Isogeometric framework.

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arXiv2023

A MEEVC discretization for two-dimensional incompressible Navier-Stokes equations with general boundary conditions

In this work, we introduce a mass, energy, enstrophy and vorticity conserving (MEEVC) mixed finite element discretization for two-dimensional incompressible Navier-Stokes equations as an alternative to the original MEEVC scheme proposed in [A. Palha and M. Gerritsma, J. Comput. Phys., 2017]. The present method can incorporate no-slip boundary conditions. Conservation properties are proven. Supportive numerical experiments with both exact and inexact quadrature are provided.

math.NA↗

arXiv2021

A mass-, kinetic energy- and helicity-conserving mimetic dual-field discretization for three-dimensional incompressible Navier-Stokes equations, part I: Periodic domains

We introduce a mimetic dual-field discretization which conserves mass, kinetic energy and helicity for three-dimensional incompressible Navier-Stokes equations. The discretization makes use of a conservative dual-field mixed weak formulation where two evolution equations of velocity are employed and dual representations of the solution are sought for each variable. A temporal discretization, which staggers the evolution equations and handles the nonlinearity such that the resulting discrete algebraic systems are linear and decoupled, is constructed. The spatial discretization is mimetic in the sense that the finite dimensional function spaces form a discrete de Rham complex. Conservation of mass, kinetic energy and helicity in the absence of dissipative terms is proven at the discrete level. Proper dissipation rates of kinetic energy and helicity in the viscous case is also proven. Numerical tests supporting the method are provided.

math.NA↗

arXiv2020

A conservative, physically compatible, dual-field discretization for turbidity currents: with application to the lock-exchange problem

In this work we present a structure preserving discretization for turbidity currents based on a mass-, energy-, enstrophy-, and vorticity-conserving formulation for 2D incompressible flows. This discretization exploits a dual-field formulation for the time evolution of the velocity and vorticity fields together with the transport equation for the particles. Due to its staggered time integration the resulting system of equations is quasi-linear, eliminating the need to solve for a fully nonlinear system of equations. It is shown that this discretization preserves the energy balance equation up to a bounded residual due to the staggering in time of the velocity and vorticity. This leads to a numerical scheme that does not introduce artificial energy dissipation. A comparison with literature results is presented showing that this approach can retrieve the dynamics of the system with a much smaller number of degrees of freedom.

math.NA↗

arXiv2020

Construction and application of algebraic dual polynomial representations for finite element methods on quadrilateral and hexahedral meshes

Given a sequence of finite element spaces which form a de Rham sequence, we will construct a dual representation of these spaces with associated differential operators which connect these spaces such that they also form a de Rham sequence. The matrix which converts primal representations to dual representations -- the Hodge matrix -- is the mass or Gram matrix. It will be shown that a bilinear form of a primal and a dual representation is equal to the vector inner product of the expansion coefficients (degrees of freedom) of both representations. This leads to very sparse system matrices, even for high order methods. The derivative of dual representations will be defined. Vector operations, grad, curl and div, for primal and dual representations are both topological and do not depend on the metric, i.e. the size and shape of the mesh or the order of the numerical method. Derivatives are evaluated by applying sparse incidence and inclusion matrices to the expansion coefficients of the representations. As illustration of the use of dual representations, the method will be applied to i) a mixed formulation for the Poisson problem in 3D, ii) it will be shown that this approach allows one to preserve the equivalence between Dirichlet and Neumann problems in the finite dimensional setting and, iii) the method will be applied to the approximation of grad-div eigenvalue problem on affine and non-affine meshes.

math.NA↗

arXiv2019

A conservative hybrid method for Darcy flow

We present a hybrid mimetic spectral element formulation for Darcy flow. The discrete representations for 1) conservation of mass, and 2) inter-element continuity, are topological relations that lead to sparse matrix systems. These constraints are independent of the element size and shape, and thus invariant under mesh transformations. The resultant algebraic system is extremely sparse even for high degree polynomial basis. Furthermore, the system can be efficiently assembled and solved for each element separately.

math.NA↗

arXiv2018

Algebraic dual polynomials for the equivalence of curl-curl problems

In this paper we will consider two curl-curl equation in two dimensions. One curl-curl problem for a scalar quantity $F$ and one problem for a vector field $\bf{E}$. For Dirichlet boundary conditions $\bf{n} \times \bf{E} =$ $ \hat{E}_{\dashv}$ on $\bf{E}$ and Neumann boundary conditions $\bf{n} \times \mbox{curl}$ $F=\hat{E}_{\dashv}$, we expect the solutions to satisfy $\bf{E}=\mbox{curl}$ $F$. When we use algebraic dual polynomial representations, these identities continue to hold at the discrete level. Equivalence will be proved and illustrated with a computational example.

math.NA↗

arXiv2018

Mimetic Spectral Element Method for Anisotropic Diffusion

This paper addresses the topological structure of steady, anisotropic, inhomogeneous diffusion problems. Two discrete formulations: a) mixed and b) direct formulations are discussed. Differential operators are represented by sparse incidence matrices, while weighted mass matrices play the role of metric-dependent Hodge matrices. The resulting mixed formulations are point-wise divergence-free if the right hand side function f = 0. The method is inf-sup stable and displays optimal convergence on orthogonal and non-affine grids.

math.NA↗

arXiv2017

Discrete conservation properties for shallow water flows using mixed mimetic spectral elements

A mixed mimetic spectral element method is applied to solve the rotating shallow water equations. The mixed method uses the recently developed spectral element histopolation functions, which exactly satisfy the fundamental theorem of calculus with respect to the standard Lagrange basis functions in one dimension. These are used to construct tensor product solution spaces which satisfy the generalized Stokes theorem, as well as the annihilation of the gradient operator by the curl and the curl by the divergence. This allows for the exact conservation of first order moments (mass, vorticity), as well as quadratic moments (energy, potential enstrophy), subject to the truncation error of the time stepping scheme. The continuity equation is solved in the strong form, such that mass conservation holds point wise, while the momentum equation is solved in the weak form such that vorticity is globally conserved. While mass, vorticity and energy conservation hold for any quadrature rule, potential enstrophy conservation is dependent on exact spatial integration. The method possesses a weak form statement of geostrophic balance due to the compatible nature of the solution spaces and arbitrarily high order spatial error convergence.

math.NA↗

arXiv2016

A mass, energy, enstrophy and vorticity conserving (MEEVC) mimetic spectral element discretization for the 2D incompressible Navier-Stokes equations

In this work we present a mimetic spectral element discretization for the 2D incompressible Navier-Stokes equations that in the limit of vanishing dissipation exactly preserves mass, kinetic energy, enstrophy and total vorticity on unstructured grids. The essential ingredients to achieve this are: (i) a velocity-vorticity formulation in rotational form, (ii) a sequence of function spaces capable of exactly satisfying the divergence free nature of the velocity field, and (iii) a conserving time integrator. Proofs for the exact discrete conservation properties are presented together with numerical test cases on highly irregular grids.

math.NA↗

arXiv2015

Mimetic spectral element method for Hamiltonian systems

There is a growing interest in the conservation of invariants when numerically solving a system of ordinary differential equations. Methods that exactly preserve these quantities in time are known as geometric integrators. In this paper we apply the recently developed mimetic framework to the solution of a system of first order ordinary differential equations. Depending on the discrete Hodge-* employed, two classes of arbitrary order time integrators are derived. It is shown that the one based on a canonical Hodge-* results in a symplectic integrator, whereas the one based on a Galerkin Hodge-* results in an energy preserving integrator. A set of numerical tests confirms these theoretical results.

math.NA↗

arXiv2013

Mixed Mimetic Spectral Element method applied to Darcy's problem

We present a discretization for Darcy's problem using the recently developed Mimetic Spectral Element Method. The gist lies in the exact discrete representation of integral relations. In this paper, an anisotropic flow through a porous medium is considered and a discretization of a full permeability tensor is presented. The performance of the method is evaluated on standard test problems, converging at the same rate as the best possible approximation.

math.NA↗