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Use of algebraic dual spaces in domain decomposition methods for Darcy flow in 3D domains

Jain, V. (author), Palha da Silva Clérigo, A. (author), Gerritsma, M.I. (author)

In this work we use algebraic dual spaces with a domain decomposition method to solve the Darcy equations. We define the broken Sobolev spaces and their finite dimensional counterparts. A global trace space is defined that connects the solution between the broken spaces. Use of algebraic dual spaces results in a sparse, metric-free...

journal article 2023

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Design of a Direct numerical Simulation of flow and heat transfer in a T-junction

Ajay Kumar, Aniketh (author), Mathur, Akshat (author), Gerritsma, M.I. (author), Komen, Ed (author)

Several investigations have been undertaken to study the velocity and temperature fields associated with the thermal mixing between fluids, and resulting thermal striping in a T-junction. However, the available experimental databases are not sufficient to describe the involved physics in adequate detail, and, due to experimental limitations,...

journal article 2023

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Efficient simulation of CO₂ migration dynamics in deep saline aquifers using a multi-task deep learning technique with consistency

Zhao, M. (author), Wang, Y. (author), Gerritsma, M.I. (author), Hajibeygi, H. (author)

CO₂ sequestration and storage in deep saline aquifers is a promising technology for mitigating the excessive concentration of the greenhouse gas in the atmosphere. However, accurately predicting the migration of CO₂ plumes requires complex multi-physics-based numerical simulation approaches, which are prohibitively...

journal article 2023

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

Zhang, Y. (author), Palha da Silva Clérigo, A. (author), Gerritsma, M.I. (author), Rebholz, Leo G. (author)

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...

journal article 2022

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Review and perspectives in quantum computing for partial differential equations in structural mechanics

Tosti Balducci, G.B.L. (author), Chen, B. Y. (author), Möller, M. (author), Gerritsma, M.I. (author), De Breuker, R. (author)

Structural mechanics is commonly modeled by (systems of) partial differential equations (PDEs). Except for very simple cases where analytical solutions exist, the use of numerical methods is required to find approximate solutions. However, for many problems of practical interest, the computational cost of classical numerical solvers running on...

review 2022

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A hybrid mimetic spectral element method for three-dimensional linear elasticity problems

Zhang, Y. (author), Fisser, Joël (author), Gerritsma, M.I. (author)

We introduce a domain decomposition structure-preserving method based on a hybrid mimetic spectral element method for three-dimensional linear elasticity problems in curvilinear conforming structured meshes. The method is an equilibrium method which satisfies pointwise equilibrium of forces. The domain decomposition is established through...

journal article 2021

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A high order hybrid mimetic discretization on curvilinear quadrilateral meshes for complex geometries

Zhang, Y. (author), Jain, V. (author), Palha da Silva Clérigo, A. (author), Gerritsma, M.I. (author)

In this paper, we present a hybrid mimetic method which solves the mixed formulation of the Poisson problem on curvilinear quadrilateral meshes. The method is hybrid in the sense that the domain is decomposed into multiple disjoint elements and the interelement continuity is enforced using a Lagrange multiplier. The method is mimetic in the...

conference paper 2020

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A conservative hybrid method for darcy flow

Jain, V. (author), Fisser, Joël (author), Palha da Silva Clérigo, A. (author), Gerritsma, M.I. (author)

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...

conference paper 2020

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

Jain, V. (author), Zhang, Y. (author), Palha da Silva Clérigo, A. (author), Gerritsma, M.I. (author)

Given a sequence of finite element spaces which form a de Rham sequence, we will construct dual representations of these spaces with associated differential operators which connect these spaces such that they also form a de Rham sequence. The dual representations also need to satisfy the de Rham sequence on the domain boundary. The matrix...

journal article 2020

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Discrete equivalence of adjoint neumann–dirichlet div-grad and grad-div equations in curvilinear 3d domains

Zhang, Yi (author), Jain, V. (author), Palha da Silva Clérigo, A. (author), Gerritsma, M.I. (author)

In this paper, we will show that the equivalence of a div-grad Neumann problem and a grad-div Dirichlet problem can be preserved at the discrete level in 3-dimensional curvilinear domains if algebraic dual polynomial representations are employed. These representations will be introduced. Proof of the equivalence at the discrete level follows...

conference paper 2020

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Spectral element method for parabolic interface problems

Khan, Arbaz (author), Upadhyay, Chandra Shekhar (author), Gerritsma, M.I. (author)

In this paper, an h∕p spectral element method with least-square formulation for parabolic interface problem will be presented. The regularity result of the parabolic interface problem is proven for non-homogeneous interface data. The differentiability estimates and the main stability estimate theorem, using non-conforming spectral element...

journal article 2018

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Spectral mimetic least-squares method for div-curl systems

Gerritsma, M.I. (author), Palha da Silva Clérigo, A. (author)

In this paper the spectral mimetic least-squares method is applied to a two-dimensional div-curl system. A test problem is solved on orthogonal and curvilinear meshes and both h- and p-convergence results are presented. The resulting solutions will be pointwise divergence-free for these test problems. For N> 1 optimal convergence rates on...

conference paper 2018

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Mimetic Least-Squares: A Least-Squares Formulation with Exact Conservation Properties

Bochev, P. (author), Gerritsma, M.I. (author)

We present a spectral mimetic least-squares method which is fully conservative and decouples the primal and dual variables.

conference paper 2014

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The geometric basis of mimetic spectral approximations

Gerritsma, M.I. (author)

lecture notes 2012

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Least-squares spectral element methods for compressible flows

Gerritsma, M.I. (author)

conference paper 2008

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Time-Dependent Polynomial Chaos

Gerritsma, M.I. (author), Vos, P. (author), Van der Steen, J.B. (author)

conference paper 2008

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Constrained least-squares methods for partial differential equations

De Maerschalck, B. (author), Gerritsma, M.I. (author)

Least-squares methods for partial differential equations are based on a norm-equivalence between the error norm and the residual norm. The resulting algebraic system of equations, which is symmetric positive definite, can also be obtained by solving a weighted collocation scheme using least-squares to solve the resulting algebraic equations....

conference paper 2006

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Least-squares spectral element method with implicit time integration for the inviscid Burgers equation

Oldenziel, G. (author), Gerritsma, M.I. (author)

This paper describes the use of the Least-Squares Spectral Element Method for non-linear hyperbolic equations. The one-dimensional inviscid Burgers equation is specifically subject of investigation. A second order backward difference method is used for time stepping. The behaviour of this formulation is examined by application to a testcase...

conference paper 2006

document

Constrained least-squares methods for partial differential equations

De Maerschalck, B. (author), Gerritsma, M.I. (author)

Least-squares methods for partial differential equations are based on a norm-equivalence between the error norm and the residual norm. The resulting algebraic system of equations, which is symmetric positive definite, can also be obtained by solving a weighted collocation scheme using least-squares to solve the resulting algebraic equations....

conference paper 2006

document

Application of Least-Squares Spectral Element Methods to Polynomial Chaos

Vos, P.E.J. (author), Gerritsma, M.I. (author)

This papers describes the use of the Least-Squares Spectral Element Method to polynomial Chaos to solve stochastic partial differential equations. The method will be described in detail and a comparison will be presented between the least-squares projection and the conventional Galerkin projection.

conference paper 2006