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

We present and analyze 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 Σ 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) [2], [3]). 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 do nothing outflow boundary condition.     

A boundary multiplier/fictitious domain method for the steady incompressible Navier-Stokes equations

Summary. We analyze the error of a fictitious-domain method with boundary Lagrange multiplier. It is applied to solve a non-homogeneous steady incompressible Navier-Stokes problem in a domain with a multiply-connected boundary. The interior mesh in the fictitious domain and the boundary mesh are independent, up to a mesh-length ratio. Received February 24, 1999 / Revised version received January 30, 2000 / Published online October 16, 2000     

A smooth fictitious domain/multiresolution method for elliptic equations on general domains

We propose a smooth fictitious domain/multiresolution method for enhancing the accuracy order in solving second order elliptic partial differential equations on general bivariate domains. We prove the existence and uniqueness of the solution of a corresponding discrete problem and a so-called interior error estimate which justifies the improved accuracy order. Numerical experiments are conducted on a Cassini oval.     

A unified three-dimensional method for vibration analysis of the frequency-dependent sandwich shallow shells with general boundary conditions

In this paper, we present an accurate three-dimensional formulation for the vibrations of the laminated and sandwich shallow shells. The sandwich structure is characterized by a thick viscoelastic core and two thin composite faces. Frequency dependent viscoelastic models are introduced in the sandwiches. Without any change in solution procedure, the formulation makes it quite easy to change the boundary conditions. The solution can be obtained by means of Rayleigh–Ritz process combined with the three-dimensional modified Fourier series which are actually assumed displacement functions. These functions, without need to meet the boundary conditions in advance, take the form of the three-dimensional Fourier series with several closed-form auxiliary functions which are supplemented to deal with the discontinuities at the boundaries in terms of displacements and its derivatives. Besides, only three assumed displacement variables are employed in the formulation which effectively reduces the computation cost. The reliability and accuracy of the method are demonstrated by numerical comparisons and examples with the constant viscoelastic models as well as the frequency dependent ones. Modal analysis and parametric studies are conducted to examine the influences of the boundary condition, dimension, lamination scheme, temperature and frequency dependence of the materials.     

The three-colour model with domain wall boundary conditions

We study the partition function for the three-colour model with domain wall boundary conditions. We express it in terms of certain special polynomials, which can be constructed recursively. Our method generalizes Kuperbergʼs proof of the alternating sign matrix theorem, replacing the six-vertex model used by Kuperberg with the eight-vertex-solid-on-solid model. As applications, we obtain some combinatorial results on three-colourings. We also conjecture an explicit formula for the free energy of the model.     

Three-coloring statistical model with domain wall boundary conditions: Functional equations

We consider the Baxter three-coloring model with boundary conditions of the domain wall type. In this case, it can be proved that the partition function satisfies some functional equations similar to the functional equations satisfied by the partition function of the six-vertex model for a special value of the crossing parameter.     

Three-coloring statistical model with domain wall boundary conditions: Trigonometric limit

We consider a nontrivial trigonometric limit of the three-coloring statistical model with the domain wall boundary conditions. In this limit, we solve the previously constructed functional equations and find a new determinant representation for the partial partition functions.     

A smoothness preserving fictitious domain method for elliptic boundary-value problems

^** Email: mommer{at}math.uu.nl We introduce a new fictitious domain method for the solutionof second-order elliptic boundary-value problems with Dirichletor Neumann boundary conditions on domains with C² boundary.The main advantage of this method is that it extends the solutionssmoothly, which leads to better performance by achieving higheraccuracy with fewer degrees of freedom. The method is basedon a least-squares interpretation of the fundamental requirementsthat the solution produced by a fictitious domain method shouldsatisfy. Careful choice of discretization techniques, togetherwith a special solution strategy, leads then to smooth solutionsof the resulting underdetermined problem. Numerical experimentsare provided which illustrate the performance and flexibilityof the approach.     

Hearing the shape of a general doubly-connected domain inR 3 with mixed boundary conditions

The basic problem in this paper is that of determining the geometry of an arbitrary doubly-connected region inR ³ with mixed boundary conditions, from the complete knowledge of the eigenvalues { _n } _n=1 for the three-dimensional Laplacian, using the asymptotic expansion of the spectral function ast0.     

A new fictitious domain method in shape optimization

The present paper is concerned with investigating the capability of the smoothness preserving fictitious domain method from Mommer (IMA J. Numer. Anal. 26:503–524, 2006) to shape optimization problems. We consider the problem of maximizing the Dirichlet energy functional in the class of all simply connected domains with fixed volume, where the state equation involves an elliptic second order differential operator with non-constant coefficients. Numerical experiments in two dimensions validate that we arrive at a fast and robust algorithm for the solution of the considered class of problems. The proposed method can be applied to three dimensional shape optimization problems.     

An approach for calculating correlation functions in the six-vertex model with domain wall boundary conditions

We address the problem of calculating correlation functions in the six-vertex model with domain wall boundary conditions by considering a particular nonlocal correlation function, called the row configuration probability. This correlation function can be used as a building block for computing various (both local and nonlocal) correlation functions in the model. We calculate the row configuration probability using the quantum inverse scattering method, giving the final result in terms of multiple integrals. We also discuss the relation to the emptiness formation probability, another nonlocal correlation function, which was previously computed using similar methods.     

Galerkin approximations with embedded boundary conditions for retarded delay differential equations

Finite-dimensional approximations are developed for retarded delay differential equations (DDEs). The DDE system is equivalently posed as an initial-boundary value problem consisting of hyperbolic partial differential equations (PDEs). By exploiting the equivalence of partial derivatives in space and time, we develop a new PDE representation for the DDEs that is devoid of boundary conditions. The resulting boundary condition-free PDEs are discretized using the Galerkin method with Legendre polynomials as the basis functions, whereupon we obtain a system of ordinary differential equations (ODEs) that is a finite-dimensional approximation of the original DDE system. We present several numerical examples comparing the solution obtained using the approximate ODEs to the direct numerical simulation of the original non-linear DDEs. Stability charts developed using our method are compared to existing results for linear DDEs. The presented results clearly demonstrate that the equivalent boundary condition-free PDE formulation accurately captures the dynamic behaviour of the original DDE system and facilitates the application of control theory developed for systems governed by ODEs.     

A stable Langevin model with diffusive-reflective boundary conditions

In this note, we consider the construction of a one-dimensional stable Langevin type process confined in the upper half-plane and submitted to diffusive-reflective boundary conditions whenever the particle position hits 0. We show that two main different regimes appear according to the values of the chosen parameters. We then use this study to construct the law of a (free) stable Langevin process conditioned to stay positive, thus extending earlier works on integrated Brownian motion. This construction further allows to obtain the exact asymptotics of the persistence probability of the integrated stable Lévy process. In addition, the paper is concluded by solving the associated trace problem in the symmetric case.     

Dissipative q-Dirac operator with general boundary conditions

In this paper, we consider the symmetric q-Dirac operator. We describe dissipative, accumulative, self-adjoint and the other extensions of such operators with general boundary conditions. We construct a self-adjoint dilation of dissipative operator. Hence, we determine the scattering matrix of dilation. Later, we construct a functional model of this operator and define its characteristic function. Finally, we prove that all root vectors of this operator are complete.     

A high order discontinuous Galerkin Nitsche method for elliptic problems with fictitious boundary

We present a discontinuous Galerkin method, based on the classical method of Nitsche, for elliptic problems with an immersed boundary representation on a structured grid. In such methods very small elements typically occur at the boundary, leading to breakdown of the discrete coercivity as well as numerical instabilities. In this work we propose a method that avoids using very small elements on the boundary by associating them to a neighboring element with a sufficiently large intersection with the domain. This construction allows us to prove the crucial inverse inequality that leads to a coercive bilinear form and as a consequence we obtain optimal order a priori error estimates. Furthermore, we prove a bound of the condition number of the stiffness matrix. All the results are valid for polynomials of arbitrary order. We also discuss the implementation of the method and present numerical examples in three dimensions.     

A note on relative oscillation theory for symplectic difference systems with general boundary conditions

We develop an analog of classical oscillation theory for discrete symplectic eigenvalue problems with general self-adjoint boundary conditions which, rather than measuring of the spectrum of one single problem, measures the difference between the spectra of two different problems. We prove formulas connecting the numbers of eigenvalues in a given interval for two symplectic eigenvalue problems with different self-adjoint boundary conditions. We derive as corollaries generalized interlacing properties of eigenvalues.     

A stabilized finite element method for a fictitious domain problem allowing small inclusions

The purpose of this work is to approximate numerically an elliptic partial differential equation posed on domains with small perforations (or inclusions). The approach is based on the fictitious domain method, and as the method's interest lies in the case in which the geometrical features are not resolved by the mesh, we propose a stabilized finite element method. The stabilization term is a simple, non‐consistent penalization that can be linked to the Barbosa‐Hughes approach. Stability and convergence are proved, and numerical results confirm the theory.