arXiv subjects

Wietse M. Boon

Publications and source records attributed to Wietse M. Boon.

At least 19 recordsLinked to original sources

arXiv2026

Linear Robin-type Domain Decomposition scheme for Mixed Formulation of Richards' Equation

In this work, we present a linear Robin-type domain decomposition scheme for solving the mixed formulation of Richards' equation (mLRDD-scheme), governing flow in variably saturated porous media. Assuming a highly heterogeneous porous medium consisting of multiple blocks/layers of distinct materials, we apply non-overlapping domain decomposition to isolate individual materials, yielding near-homogeneous conditions within each subdomain. Richards' equation is discretized in time by backward Euler and linearized using the L-scheme. A Robin-type interface condition, consistent with the mixed formulation, is then derived to couple the subdomains. For spatial discretization, mixed finite elements are employed, ensuring local mass conservation and providing the flux variables consistent with the Robin condition. The convergence of the proposed scheme is rigorously proved. Heterogeneous, multidomain numerical examples in 2D/3D are presented to demonstrate the efficiency and robustness of the scheme, along with numerical validation of the theoretical convergence results.

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arXiv2026

Parameter-robust Preconditioners for the Stokes-Darcy Coupled Problem without Fractional Operators

We consider the Stokes-Darcy coupled problem, which models the interaction between free-flow and porous medium flow. By enforcing the normal flux continuity interface condition directly within the finite-element spaces, we establish unified well-posedness results for the coupled system under various boundary condition scenarios. Using the operator preconditioning framework, we develop a parameter-robust preconditioner that avoids the use of fractional operators. Numerical experiments employing both $H(\operatorname{div})$-conforming and nonconforming finite-element methods are presented to confirm the theoretical findings and demonstrate the robustness of the proposed block preconditioners with respect to the physical parameters and mesh size.

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arXiv2026

Solvers for mixed finite element problems using Poincaré operators based on spanning trees

We propose a decomposition of Hilbert complexes that directly leads to a Poincaré operator. An explicit example is provided that decomposes a finite element differential complex using spanning trees in the grid. The Poincaré operator has three implications. First, it yields a new basis in which the mixed formulation of the Hodge-Laplace problem unravels from a large saddle point system into seven smaller, symmetric positive definite systems. These systems can be solved sequentially, and three of these have the same dimensions as the cohomology classes. Second, we use the operator to construct an explicit basis for the harmonic forms. Third, we propose an auxiliary space preconditioner for problems in weighted Sobolev spaces, that robustly handles the large kernel of the differential operator. These three implications are validated through numerical experiments.

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arXiv2026

H(curl)-based approximation of the Stokes problem with slip boundary conditions

Reformulating the incompressible Stokes equations with the velocity sought in H(curl) has recently emerged as a promising approach for the design of helicity-preserving schemes in magnetohydrodynamics and pressure-robust finite element methods on polygonal meshes. A key challenge in this setting, however, is the treatment of Navier slip boundary conditions. In this paper, we overcome this difficulty by recasting the slip condition as a Robin boundary condition and proving well-posedness of the resulting continuous problem. We further identify the geometric and regularity assumptions on the domain and the exact solution under which the classical Stokes solution is recovered. Finally, we study a conforming finite element Galerkin discretization, establishing stability and a priori error estimates. Numerical experiments validate the optimal convergence rates predicted by the theory.

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arXiv2025

Multipoint stress mixed finite element methods for the linear Cosserat equations

We propose mixed finite element methods for Cosserat materials that use suitable quadrature rules to eliminate the Cauchy and coupled stress variables locally. The reduced system consists of only the displacement and rotation variables. Four variants are proposed for which we show stability and convergence using a priori estimates. Numerical experiments verify the theoretical findings and higher order convergence is observed in some variables.

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arXiv2025

Solving Biot poroelasticity by coupling OPM Flow with the two-point stress approximation finite volume method

Finite volume methods are prevalent in reservoir simulation due to their mass conservation properties and their ability to handle complex grids. However, a simple and consistent finite volume method for elasticity was unavailable until the recently developed two-point stress approximation finite volume method (TPSA). In this work, we show how to couple TPSA to an established flow simulator, using OPM Flow as our primary example. Due to this choice of numerical methods, the coupling is naturally handled at the cell centers, without requiring interpolation operators. We propose a fixed stress coupling scheme and reuse algebraic multi-grid preconditioners, which are known to be effective for two-point flux finite volume methods. Numerical examples illustrate the flexibility of the approach and we showcase how the introduction of solid mechanics impacts the behavior of compartmentalized flow systems.

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arXiv2025

H(curl)-based approximation of the Stokes problem with weakly enforced no-slip boundary conditions

In this work, we show how to impose no-slip boundary conditions for an H(curl)-based formulation for incompressible Stokes flow, which is used in structure-preserving discretizations of Navier-Stokes and magnetohydrodynamics equations. At first glance, it seems straightforward to apply no-slip boundary conditions: the tangential part is an essential boundary condition on H(curl) and the normal component can be naturally enforced through integration-by-parts of the divergence term. However, we show that this can lead to an ill-posed discretization and propose a Nitsche-based finite element method instead. We analyze the discrete system, establishing stability and deriving a priori error estimates. Numerical experiments validate our analysis and demonstrate optimal convergence rates for the velocity field.

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arXiv2025

Fitted norm preconditioners for the Hodge Laplacian in mixed form

We use the practical framework for abstract perturbed saddle point problems recently introduced by Hong et al. to analyze the mixed formulation of the Hodge Laplace problem. We compose two parameter-dependent norms in which the uniform continuity and stability of the problem follow. This not only guarantees the well-posedness of the corresponding variational formulation on the continuous level, but also of related compatible discrete models. We further simplify the obtained norms and, in both cases, arrive at the same norm-equivalent preconditioner that is easily implementable. The efficiency and uniformity of the preconditioner are demonstrated numerically by the fast convergence and uniformly bounded number of preconditioned MINRES iterations required to solve various instances of Hodge Laplace problems in two and three space dimensions.

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arXiv2024

Neural network solvers for parametrized elasticity problems that conserve linear and angular momentum

We consider a mixed formulation of parametrized elasticity problems in terms of stress, displacement, and rotation. The latter two variables act as Lagrange multipliers to enforce conservation of linear and angular momentum. Due to the saddle-point structure, the resulting system is computationally demanding to solve directly, and we therefore propose an efficient solution strategy based on a decomposition of the stress variable. First, a triangular system is solved to obtain a stress field that balances the body and boundary forces. Second, a trained neural network is employed to provide a correction without affecting the conservation equations. The displacement and rotation can be obtained by post-processing, if necessary. The potential of the approach is highlighted by three numerical test cases, including a non-linear model.

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arXiv2024

Mixed finite element and TPSA finite volume methods for linearized elasticity and Cosserat materials

Cosserat theory of elasticity is a generalization of classical elasticity that allows for asymmetry in the stress tensor by taking into account micropolar rotations in the medium. The equations involve a rotation field and associated "couple stress" as variables, in addition to the conventional displacement and Cauchy stress fields. In recent work, we derived a mixed finite element method (MFEM) for the linear Cosserat equations that converges optimally in these four variables. The drawback of this method is that it retains the stresses as unknowns, and therefore leads to relatively large saddle point system that are computationally demanding to solve. As an alternative, we developed a finite volume method in which the stress variables are approximated using a minimal, two-point stencil (TPSA). The system consists of the displacement and rotation variables, with an additional "solid pressure" unknown. Both the MFEM and TPSA methods are robust in the incompressible limit and in the Cauchy limit, for which the Cosserat equations degenerate to classical linearized elasticity. We report on the construction of the methods, their a priori properties, and compare their numerical performance against an MPSA finite volume method.

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arXiv2024

A mortar method for the coupled Stokes-Darcy problem using the MAC scheme for Stokes and mixed finite elements for Darcy

A discretization method with non-matching grids is proposed for the coupled Stokes-Darcy problem that uses a mortar variable at the interface to couple the marker and cell (MAC) method in the Stokes domain with the Raviart-Thomas mixed finite element pair in the Darcy domain. Due to this choice, the method conserves linear momentum and mass locally in the Stokes domain and exhibits local mass conservation in the Darcy domain. The MAC scheme is reformulated as a mixed finite element method on a staggered grid, which allows for the proposed scheme to be analyzed as a mortar mixed finite element method. We show that the discrete system is well-posed and derive a priori error estimates that indicate first order convergence in all variables. The system can be reduced to an interface problem concerning only the mortar variables, leading to a non-overlapping domain decomposition method. Numerical examples are presented to illustrate the theoretical results and the applicability of the method.

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arXiv2023

Deep learning based reduced order modeling of Darcy flow systems with local mass conservation

We propose a new reduced order modeling strategy for tackling parametrized Partial Differential Equations (PDEs) with linear constraints, in particular Darcy flow systems in which the constraint is given by mass conservation. Our approach employs classical neural network architectures and supervised learning, but it is constructed in such a way that the resulting Reduced Order Model (ROM) is guaranteed to satisfy the linear constraints exactly. The procedure is based on a splitting of the PDE solution into a particular solution satisfying the constraint and a homogenous solution. The homogeneous solution is approximated by mapping a suitable potential function, generated by a neural network model, onto the kernel of the constraint operator; for the particular solution, instead, we propose an efficient spanning tree algorithm. Starting from this paradigm, we present three approaches that follow this methodology, obtained by exploring different choices of the potential spaces: from empirical ones, derived via Proper Orthogonal Decomposition (POD), to more abstract ones based on differential complexes. All proposed approaches combine computational efficiency with rigorous mathematical interpretation, thus guaranteeing the explainability of the model outputs. To demonstrate the efficacy of the proposed strategies and to emphasize their advantages over vanilla black-box approaches, we present a series of numerical experiments on fluid flows in porous media, ranging from mixed-dimensional problems to nonlinear systems. This research lays the foundation for further exploration and development in the realm of model order reduction, potentially unlocking new capabilities and solutions in computational geosciences and beyond.

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arXiv2022

Mixed and multipoint finite element methods for rotation-based poroelasticity

This work proposes a mixed finite element method for the Biot poroelasticity equations that employs the lowest-order Raviart-Thomas finite element space for the solid displacement and piecewise constants for the fluid pressure. The method is based on the formulation of linearized elasticity as a weighted vector Laplace problem. By introducing the solid rotation and fluid flux as auxiliary variables, we form a four-field formulation of the Biot system, which is discretized using conforming mixed finite element spaces. The auxiliary variables are subsequently removed from the system in a local hybridization technique to obtain a multipoint rotation-flux mixed finite element method. Stability and convergence of the four-field and multipoint mixed finite element methods are shown in terms of weighted norms, which additionally leads to parameter-robust preconditioners. Numerical experiments confirm the theoretical results.

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arXiv2022

Flux-mortar mixed finite element methods with multipoint flux approximation

The flux-mortar mixed finite element method was recently developed for a general class of domain decomposition saddle point problems on non-matching grids. In this work we develop the method for Darcy flow using the multipoint flux approximation as the subdomain discretization. The subdomain problems involve solving positive definite cell-centered pressure systems. The normal flux on the subdomain interfaces is the mortar coupling variable, which plays the role of a Lagrange multiplier to impose weakly continuity of pressure. We present well-posedness and error analysis based on reformulating the method as a mixed finite element method with a quadrature rule. We develop a non-overlapping domain decomposition algorithm for the solution of the resulting algebraic system that reduces it to an interface problem for the flux-mortar, as well as an efficient interface preconditioner. A series of numerical experiments is presented illustrating the performance of the method on general grids, including applications to flow in complex porous media.

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arXiv2022

A Parameter-Robust Iterative Method for Stokes-Darcy Problems Retaining Local Mass Conservation

We consider a coupled model of free-flow and porous medium flow, governed by stationary Stokes and Darcy flow, respectively. The coupling between the two systems is enforced by introducing a single variable representing the normal flux across the interface. The problem is reduced to a system concerning only the interface flux variable, which is shown to be well-posed in appropriately weighted norms. An iterative solution scheme is then proposed to solve the reduced problem such that mass is conserved at each iteration. By introducing a preconditioner based on the weighted norms from the analysis, the performance of the iterative scheme is shown to be robust with respect to material and discretization parameters. By construction, the scheme is applicable to a wide range of locally conservative discretization schemes and we consider explicit examples in the framework of Mixed Finite Element methods. Finally, the theoretical results are confirmed with the use of numerical experiments.

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arXiv2022

A multipoint vorticity mixed finite element method for incompressible Stokes flow

We propose a mixed finite element method for Stokes flow with one degree of freedom per element and facet of simplicial grids. The method is derived by considering the vorticity-velocity-pressure formulation and eliminating the vorticity locally through the use of a quadrature rule. The discrete solution is pointwise divergence-free and the method is pressure robust. The theoretically derived convergence rates are confirmed by numerical experiments.

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arXiv2022

A Reduced Basis Method for Darcy flow systems that ensures local mass conservation by using exact discrete complexes

A solution technique is proposed for flows in porous media that guarantees local conservation of mass. We first compute a flux field to balance the mass source and then exploit exact co-chain complexes to generate a solenoidal correction. A reduced basis method based on proper orthogonal decomposition is employed to construct the correction and we show that mass balance is ensured regardless of the quality of the reduced basis approximation. The method is directly applicable to mixed finite and virtual element methods, among other structure-preserving discretization techniques, and we present the extension to Darcy flow in fractured porous media.

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arXiv2022

An Adaptive Penalty Method for Inequality Constrained Minimization Problems

The primal-dual active set method is observed to be the limit of a sequence of penalty formulations. Using this perspective, we propose a penalty method that adaptively becomes the active set method as the residual of the iterate decreases. The adaptive penalty method (APM) therewith combines the main advantages of both methods, namely the ease of implementation of penalty methods and the exact imposition of inequality constraints inherent to the active set method. The scheme can be considered a quasi-Newton method in which the Jacobian is approximated using a penalty parameter. This spatially varying parameter is chosen at each iteration by solving an auxiliary problem.

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