Stokes Interface Problem Research Articles

The weak Galerkin finite element method for Stokes interface problems with curved interface

In this paper, we develop a weak Galerkin (WG) finite element scheme for the Stokes interface problems with curved interface. The conventional numerical schemes rely on the use of straight segments to approximate the curved interface and the accuracy is limited by geometric errors. Hence in our method, we directly construct the weak Galerkin finite element space on the curved cells to avoid geometric errors. For the integral calculation on curved cells, we employ non-affine transformations to map curved cells onto the reference element. The optimal error estimates are obtained in both the energy norm and the L2 norm. A series of numerical experiments are provided to validate the efficiency of the proposed WG method.

A Hybrid Neural-Network and MAC Scheme for Stokes Interface Problems

A Hybrid Neural-Network and MAC Scheme for Stokes Interface Problems

Analysis and Computation of a Weak Galerkin Scheme for Solving the 2D/3D Stationary Stokes Interface Problems with High-Order Elements

Abstract In this paper, we present a high-order weak Galerkin finite element method (WG-FEM) for solving the stationary Stokes interface problems with discontinuous velocity and pressure in ℝ d (d = 2, 3). This WG method is equipped with stable finite elements consisting of usual polynomials of degree k ≥ 1 for the velocity and polynomials of degree k – 1 for the pressure, both are discontinuous. Optimal convergence rates of order k + 1 for the velocity and order k for the pressure are established in L 2-norm on hybrid meshes. Numerical experiments verify the expected order of accuracy for both two-dimensional and three-dimensional examples. Moreover, numerically it is shown that the proposed WG algorithm is able to accommodate geometrically complicated and very irregular interfaces having sharp edges, cusps, and tips.

A Petrov–Galerkin immersed finite element method for steady Navier–Stokes interface problem with non-homogeneous jump conditions

This paper develops a lowest-order Petrov–Galerkin immersed Q1−Q0 method for solving Stokes interface problem. We make use of a uniform, interface-unfitted Cartesian mesh. An immersed Petrov–Galerkin formulation is presented, where the test spaces are conventional finite element spaces and the solution spaces satisfying the jump conditions. For the Stokes and Navier–Stokes interface problem, simple stabilized items are introduced. The nonlinear convective term is treated using Picard’s iteration. Extensive numerical experiments validate the feasibility and optimal convergence order for the Petrov–Galerkin immersed Q1−Q0 scheme both with homogeneous and non-homogeneous jumps.

The random feature method for solving interface problems

Interface problems have long been a major focus of scientific computing, leading to the development of various numerical methods. Traditional mesh-based methods often employ time-consuming body-fitted meshes with standard discretization schemes or unfitted meshes with tailored schemes to achieve controllable accuracy and convergence rate. Along another line, mesh-free methods bypass mesh generation but lack robustness in terms of convergence and accuracy due to the low regularity of solutions. In this study, we propose a novel method for solving interface problems within the framework of the random feature method (RFM). This approach utilizes random feature functions in conjunction with a partition of unity as approximation functions, and solves a linear least-squares system to obtain the approximate solution. In the context of interface problems, two innovative and crucial components are incorporated into the RFM. Firstly, we utilize two sets of random feature functions on each side of the interface, allowing for the inclusion of low regularity or even discontinuous behaviors in the solution. Secondly, the construction of the loss function is based on the assessment of the partial differential equation, initial/boundary conditions, and the interface condition on collocation points. This approach ensures that these conditions are equally satisfied. Consequently, the challenges arising from geometric complexity primarily manifest in the generation of collocation points, a task amenable to standard methods. Importantly, the proposed method retains its meshfree characteristics and robustness when addressing problems featuring intricate geometries. We validate our method through a series of linear interface problems with increasingly complex geometries, including two-dimensional elliptic and three-dimensional Stokes interface problems, a three-dimensional elasticity interface problem, a moving interface problem with topological change, a dynamic interface problem with large deformation, and a linear fluid–solid interaction problem with complex geometry. Our findings show that despite the solution often being only continuous or even discontinuous, our method not only eliminates the need for mesh generation but also maintains high accuracy, akin to the spectral collocation method for smooth solutions. Remarkably, for the same accuracy requirement, our method requires two to three orders of magnitude fewer degrees of freedom than traditional methods, demonstrating its significant potential for solving interface problems with complex geometries and predetermined intricate evolution.

Second Order Convergence of a Modified MAC Scheme for Stokes Interface Problems

Stokes flow equations have been implemented successfully in practice for simulating problems with moving interfaces. Though computational methods based on the well-known MAC scheme produce accurate solutions and numerical convergence can be demonstrated using a resolution study, the rigorous convergence proofs are usually limited to particular reformulations and boundary conditions. In this paper, a rigorous error analysis of the marker and cell scheme for Stokes interface problems with constant viscosity in the framework of the finite difference method is presented. Without reformulating the problem into elliptic PDEs, the main idea is to use a discrete Ladyzenskaja-Babuska-Brezzi condition and construct auxiliary functions, which satisfy discretized Stokes equations and possess at least second order accuracy in the neighborhood of the moving interface. In particular, the method, for the first time, enables one to prove second order convergence of the velocity gradient in the discrete $$\ell ^2$$ -norm, in addition to the velocity and pressure fields. Numerical experiments verify the desired properties of the methods and the expected order of accuracy for both two-dimensional and three-dimensional examples.

A Discontinuity and Cusp Capturing PINN for Stokes Interface Problems with Discontinuous Viscosity and Singular Forces

A Discontinuity and Cusp Capturing PINN for Stokes Interface Problems with Discontinuous Viscosity and Singular Forces

A face‐centred finite volume method for high‐contrast Stokes interface problems

AbstractA new face‐centred finite volume method (FCFV) for Stokes problems involving sharp interfaces is proposed. Two formulations, based on two strong forms of the Stokes problem, and using different mixed variables, are presented. Particular attention is paid to the symmetry of the resulting system of global equations, and a simple rewriting of the interface boundary condition is proposed to ensure that one of the formulations preserves the symmetry of the linear system that is usually lost when considering material interfaces. Four numerical examples are considered to test the implementation numerically by performing mesh convergence studies, in two and three dimensions. The examples account for discontinuous viscosity as well as the effect of surface tension. The results show that one of the formulations is less sensitive to the numerical stabilisation used in FCFV methods but does not preserve the symmetry of the global system, whereas the other formulation is more sensitive to the stabilisation, but preserves the symmetry of the resulting system of equations. The FCFV method appears as a promising alternative for the simulation of viscous flow involving internal boundaries on conformal meshes. The potential application of the FCFV method for the purpose of geodynamic modelling is discussed.

An immersed [formula omitted]-[formula omitted] element for Stokes interface problems and the optimal convergence analysis

This paper presents and analyzes an immersed finite element (IFE) method for solving Stokes interface problems with a piecewise constant viscosity coefficient that has a jump across the interface. In the method, the triangulation does not need to fit the interface and the IFE spaces are constructed from the traditional CR−P0 element with modifications near the interface according to the interface jump conditions. We prove that the IFE basis functions are unisolvent on arbitrary triangles without any angle conditions and the IFE spaces have the optimal approximation capabilities, although the proof is challenging due to the coupling of the velocity and the pressure. The stability and the optimal error estimates of the proposed IFE method are also derived rigorously. The constants in the error estimates are shown to be independent of the interface location relative to the triangulation. Numerical examples are provided to verify the theoretical results.

A stabilized nonconforming Nitsche's extended finite element method for Stokes interface problems

<p style='text-indent:20px;'>In this paper, a stabilized extended finite element method is proposed for Stokes interface problems on unfitted triangulation elements which do not require the interface align with the triangulation. The problem is written on mixed form using nonconforming <inline-formula><tex-math id="M1">\begin{document}$ P_1 $\end{document}</tex-math></inline-formula> velocity and elementwise <inline-formula><tex-math id="M2">\begin{document}$ P_0 $\end{document}</tex-math></inline-formula> pressure. Extra stabilization terms involving velocity and pressure are added in the discrete bilinear form. An inf-sup stability result is derived, which is uniform with respect to mesh size <inline-formula><tex-math id="M3">\begin{document}$ h $\end{document}</tex-math></inline-formula>, the viscosity and the position of the interface. An optimal priori error estimates are obtained. Moreover, the errors in energy norm for velocity and in <inline-formula><tex-math id="M4">\begin{document}$ L^2 $\end{document}</tex-math></inline-formula> norm for pressure are uniform to the viscosity and the location of the interface. Results of numerical experiments are presented to support the theoretical analysis.</p>

A generalized finite difference method for solving Stokes interface problems

In this paper, a new scheme is proposed to solve the Stokes interface problem. The scheme turns the Stokes interface problem into two coupled Stokes non-interface subproblems and adds a mixed boundary condition to overcome the numerical pressure oscillation. Since the interface becomes the boundary of the subproblems, the scheme has the advantage to deal with the interface problem with complex geometry. Furthermore, a generalized finite difference method (GFDM) is adopted to solve the coupled Stokes non-interface subproblems. The GFDM is developed from the Taylor series expansions and moving-least squares approximation. Due to the flexibility of the GFDM, it is convenient to handle the complex boundary conditions that appeared in the proposed scheme. The numerical examples verify the accuracy and stability of the GFDM to solve the Stokes interface problem with the mixed boundary conditions. Moreover, for some given numerical examples, the proposed scheme is more accurate than the classical formula of the pressure Poisson equation, especially in terms of pressure.

A class of nonconforming immersed finite element methods for Stokes interface problems

In this paper, we introduce a class of lowest-order nonconforming immersed finite element (IFE) methods for solving two-dimensional Stokes interface problems. The proposed methods do not require the solution mesh to align with the fluid interface and can use either triangular or rectangular meshes. On triangular meshes, the Crouzeix–Raviart element is used for velocity approximation, and piecewise constant for pressure. On rectangular meshes, the Rannacher–Turek rotated Q1-Q0 finite element is used. The new vector-valued IFE functions are constructed to approximate the interface jump conditions. Basic properties including the unisolvency and the partition of unity of these new IFE functions are discussed. Approximation capabilities of the new IFE spaces for the Stokes interface problems are examined through a series of numerical examples. Numerical approximations in the L2-norm and the broken H1-norm for the velocity and the L2-norm for the pressure are observed to converge optimally.

A conforming-nonconforming mixed immersed finite element method for unsteady Stokes equations with moving interfaces

<p style='text-indent:20px;'>In this article, we develop a new mixed immersed finite element discretization for two-dimensional unsteady Stokes interface problems with unfitted meshes. The proposed IFE spaces use conforming linear elements for one velocity component and non-conforming linear elements for the other velocity component. The pressure is approximated by piecewise constant. Unisolvency, among other fundamental properties of the new vector-valued IFE functions, is analyzed. Based on the new IFE spaces, semi-discrete and full-discrete schemes are developed for solving the unsteady Stokes equations with a stationary or a moving interface. Re-meshing is not required in our numerical scheme for solving the moving-interface problem. Numerical experiments are carried out to demonstrate the performance of this new IFE method.</p>

An unfitted hybrid high-order method for the Stokes interface problem

Abstract We design and analyze a hybrid high-order method on unfitted meshes to approximate the Stokes interface problem. The interface can cut through the mesh cells in a very general fashion. A cell-agglomeration procedure prevents the appearance of small cut cells. Our main results are inf-sup stability and a priori error estimates with optimal convergence rates in the energy norm. Numerical simulations corroborate these results.

A Nonconforming Nitsche’s Extended Finite Element Method for Stokes Interface Problems

This article studies the nonconforming $$P_{1}/P_{0}$$ Nitsche’s extended finite element method for Stokes interface problems with interface-unfitted meshes. We derive the inf-sup stability result and optimal a priori error estimates in spite of the low regularity of interface problems. It is shown that all results are independent of not only the viscosity parameters but also the position of the interface without other assumption for the interface. Numerical experiments are carried out to validate theoretical results.

Least squares spectral method for the two-dimensional Stokes interface problems

This paper deals with the pseudo-spectral least squares method to approximate the solution of the Stokes interface problem, that is, the Stokes equations with discontinuous viscosity and singular source term. The Stokes interface equations are adjusted in each sub-domain along with the interface conditions. Then the equations are reformulated as a first order system by introducing stress tensor as a new independent variable. The continuous and discrete norm least squares functionals using Legendre and Chebyshev weights for the first order system are defined and shown to be equivalent to appropriate product norms. The spectral convergence of the proposed method is presented. Numerical examples are also given to demonstrate the method and analysis.

Distributed Lagrange Multiplier/Fictitious Domain Finite Element Method for a Transient Stokes Interface Problem with Jump Coefficients

The distributed Lagrange multiplier/fictitious domain (DLM/FD)-mixed finite element method is developed and analyzed in this paper for a transient Stokes interface problem with jump coefficients. The semi- and fully discrete DLM/FD-mixed finite element scheme are developed for the first time for this problem with a moving interface, where the arbitrary Lagrangian-Eulerian (ALE) technique is employed to deal with the moving and immersed subdomain. Stability and optimal convergence properties are obtained for both schemes. Numerical experiments are carried out for different scenarios of jump coefficients, and all theoretical results are validated.

A time dependent Stokes interface problem: well-posedness and space-time finite element discretization

In this paper a time dependent Stokes problem that is motivated by a standard sharp interface model for the fluid dynamics of two-phase flows is studied. This Stokes interface problem has discontinuous density and viscosity coefficients and a pressure solution that is discontinuous across an evolving interface. This strongly simplified two-phase Stokes equation is considered to be a good model problem for the development and analysis of finite element discretization methods for two-phase flow problems. In view of theunfitted finite element methods that are often used for two-phase flow simulations, we are particularly interested in a well-posed variational formulation of this Stokes interface problem in a Euclidean setting. Such well-posed weak formulations, which are not known in the literature, are the main results of this paper. Different variants are considered, namely one with suitable spaces of divergence free functions, a discrete-in-time version of it, and variants in which the divergence free constraint in the solution space is treated by a pressure Lagrange multiplier. The discrete-in-time variational formulation involving the pressure variable for the divergence free constraint is a natural starting point for a space-time finite element discretization. Such a method is introduced and results of numerical experiments with this method are presented.

A conforming enriched finite element method for Stokes interface problems

A new conforming enriched finite element method is presented for Stokes interface problems with interface-unfitted meshes. The conforming enriched finite element pair is constructed based on the MINI element pair. A ghost penalty term is used in the standard discretization form as a stabilization term. An inf–sup stability result is derived, which is uniform with respect to the mesh size. Our method does not limit the viscosity coefficient to be a piecewise constant. Finite element errors are proved to be optimal. Numerical experiments are carried out to validate theoretical results.

Inf-sup stability of geometrically unfitted Stokes finite elements

The paper shows an inf-sup stability property for several well-known 2D and 3D Stokes elements on triangulations which are not fitted to a given smooth or polygonal domain. The property implies stability and optimal error estimates for a class of unfitted finite element methods for the Stokes and Stokes interface problems, such as Nitsche-XFEM or cutFEM. The error analysis is presented for the Stokes problem. All assumptions made in the paper are satisfied once the background mesh is shape-regular and fine enough.