The algebraic immersed interface and boundary method for elliptic equations with jump conditions

A new fictitious domain method, the algebraic immersed interface and boundary (AIIB) method, is presented for elliptic equations with immersed interface conditions. This method allows jump conditions on immersed interfaces to be discretized accurately. The main idea is to create auxiliary unknowns at existing grid locations which increases the degrees of freedom of the initial problem. These auxiliary unknowns allow to impose various constraints to the system on interfaces of complex shapes. For instance, the method is able to deal with immersed interfaces for elliptic equations with jump conditions on the solution or discontinuous coefficients with a second order of spatial accuracy. As the AIIB method acts on an algebraic level and only changes the problem matrix, no particular attention to the initial discretization is required. The method can be easily implemented in any structured grid code and can deal with immersed boundary problems too. Several validation problems are presented to demonstrate the interest and accuracy of the method

An efficient neural-network and finite-difference hybrid method for elliptic interface problems with applications

A new and efficient neural-network and finite-difference hybrid method is developed for solving Poisson equation in a regular domain with jump discontinuities on embedded irregular interfaces. Since the solution has low regularity across the interface, when applying finite difference discretization to this problem, an additional treatment accounting for the jump discontinuities must be employed. Here, we aim to elevate such an extra effort to ease our implementation by machine learning methodology. The key idea is to decompose the solution into singular and regular parts. The neural network learning machinery incorporating the given jump conditions finds the singular solution, while the standard finite difference method is used to obtain the regular solution with associated boundary conditions. Regardless of the interface geometry, these two tasks only require supervised learning for function approximation and a fast direct solver for Poisson equation, making the hybrid method easy to implement and efficient. The two- and three-dimensional numerical results show that the present hybrid method preserves second-order accuracy for the solution and its derivatives, and it is comparable with the traditional immersed interface method in the literature. As an application, we solve the Stokes equations with singular forces to demonstrate the robustness of the present method

A fictitious domain model for the Stokes/Brinkman problem with jump embedded boundary conditions

International audienceWe present and analyse a new fictitious domain model for the Brinkman or Stokes/Brinkman problems in order to handle general jump embedded boundary conditions (J.E.B.C.) on an immersed interface. Our model is based on algebraic transmission conditions combining the stress and velocity jumps on the interface $\S$ separating two subdomains: they are well chosen to get the coercivity of the operator. It is issued from a generalization to vector elliptic problems of a previous model stated for scalar problems with jump boundary conditions (Angot (2003, 2005) \cite{Ang03,Ang05}). The proposed model is first proved to be well-posed in the whole fictitious domain and some sub-models are identified. A family of fictitious domain methods can be then derived within the same unified formulation which provides various interface or boundary conditions, e.g. a given stress of Neumann or Fourier type or a velocity Dirichlet condition. In particular, we prove the consistency of the given-traction E.B.C. method including the so-called {\em do nothing} outflow boundary condition

An Immersed Interface Method for Discrete Surfaces

Fluid-structure systems occur in a range of scientific and engineering applications. The immersed boundary(IB) method is a widely recognized and effective modeling paradigm for simulating fluid-structure interaction(FSI) in such systems, but a difficulty of the IB formulation is that the pressure and viscous stress are generally discontinuous at the interface. The conventional IB method regularizes these discontinuities, which typically yields low-order accuracy at these interfaces. The immersed interface method(IIM) is an IB-like approach to FSI that sharply imposes stress jump conditions, enabling higher-order accuracy, but prior applications of the IIM have been largely restricted to methods that rely on smooth representations of the interface geometry. This paper introduces an IIM that uses only a C0 representation of the interface,such as those provided by standard nodal Lagrangian FE methods. Verification examples for models with prescribed motion demonstrate that the method sharply resolves stress discontinuities along the IB while avoiding the need for analytic information of the interface geometry. We demonstrate that only the lowest-order jump conditions for the pressure and velocity gradient are required to realize global 2nd-order accuracy. Specifically,we show 2nd-order global convergence rate along with nearly 2nd-order local convergence in the Eulerian velocity, and between 1st-and 2nd-order global convergence rates along with 1st-order local convergence for the Eulerian pressure. We also show 2nd-order local convergence in the interfacial displacement and velocity along with 1st-order local convergence in the fluid traction. As a demonstration of the method's ability to tackle complex geometries,this approach is also used to simulate flow in an anatomical model of the inferior vena cava.Comment: - Added a non-axisymmetric example (flow within eccentric rotating cylinder in Sec. 4.3) - Added a more in-depth analysis and comparison with a body-fitted approach for the application in Sec. 4.

A class of immersed finite element methods for Stokes interface problems

In this dissertation, we explore applications of partial differential equations with discontinuous coefficients. We consider the nonconforming immersed finite element methods (IFE) for modeling and simulating these partial differential equations. A one-dimensional second-order parabolic initial-boundary value problem with discontinuous coefficients is studied. We propose an extension of the immersed finite element method to a high-order immersed finite element method for solving one-dimensional parabolic interface problems. In addition, we introduce a nonconforming immersed finite element method to solve the two-dimensional parabolic problem with a moving interface. In the nonconforming IFE framework, the degrees of freedom are determined by the average integral value over the element edges. The continuity of the nonconforming IFE framework is in the weak sense in comparison the continuity of the conforming IFE framework. Numerical experiments are provided to demonstrate the features and the robustness of these methods. We introduce a class of lowest-order nonconforming immersed finite element methods for solving two-dimensional Stokes interface problem. On triangular meshes, the Crouzeix-Raviart element is used for velocity approximation, and piecewise constant for pressure. On rectangular meshes, the Rannacher-Turek rotated $Q_1$-$Q_0$ finite element is used. We also consider a new mixed immersed finite element method for the Stokes interface problem on an unfitted mesh. The proposed IFE space uses conforming linear elements for one velocity component and nonconforming linear elements for the other component. 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 methods are discussed. Numerical approximations are observed to converge optimally. Lastly, we apply each class of the new immersed finite element methods to solve the unsteady Stokes interface problem. 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. A comparison of the degrees of freedom and number of elements are presented for each method. Numerical experiments are provided to demonstrate the features of these methods