A method of graph transformation type for numerical simulation of invariant manifolds

We consider the problem of numerical projection onto local stable and local unstable manifolds defined in a neighborhood of a hyperbolic-type trajectory. We propose effective iteration algorithms, prove their convergence, and present the results of numerical calculations for two-dimensional equations of Navier-Stokes type.     

A Two-Phase Algorithm for a Variational Inequality Formulation of Equilibrium Problems

We introduce an explicit algorithm for solving nonsmooth equilibrium problems in finite-dimensional spaces. A particular iteration proceeds in two phases. In the first phase, an orthogonal projection onto the feasible set is replaced by projections onto suitable hyperplanes. In the second phase, a projected subgradient type iteration is replaced by a specific projection onto a halfspace. We prove, under suitable assumptions, convergence of the whole generated sequence to a solution of the problem. The proposed algorithm has a low computational cost per iteration and, some numerical results are reported.     

Algorithms for separable nonlinear least squares with application to modelling time-resolved spectra

The multiexponential analysis problem of fitting kinetic models to time-resolved spectra is often solved using gradient-based algorithms that treat the spectral parameters as conditionally linear. We make a comparison of the two most-applied such algorithms, alternating least squares and variable projection. A numerical study examines computational efficiency and linear approximation standard error estimates. A new derivation of the Fisher information matrix under the full Golub-Pereyra gradient allows a numerical comparison of parameter precision under variable projection variants. Under the criteria of efficiency, quality of standard error estimates and parameter precision, we conclude that the Kaufman variable projection technique performs well, while techniques based on alternating least squares have significant disadvantages for application in the problem domain.     

Finite element analysis of parabolic integro-differential equations of Kirchhoff type

The aim of this paper is to study parabolic integro-differential equations of Kirchhoff type. We prove the existence and uniqueness of the solution for this problem via Galerkin method. Semidiscrete formulation for this problem is presented using conforming finite element method. As a consequence of the Ritz–Volterra projection, we derive error estimates for both semidiscrete solution and its time derivative. To find the numerical solution of this class of equations, we develop two different types of numerical schemes, which are based on backward Euler–Galerkin method and Crank–Nicolson–Galerkin method. A priori bounds and convergence estimates in spatial as well as temporal direction of the proposed schemes are established. Finally, we conclude this work by implementing some numerical experiments to confirm our theoretical results.     

Numerical Computation of Connecting Orbits in Delay Differential Equations

We discuss the numerical computation of homoclinic and heteroclinic orbits in delay differential equations. Such connecting orbits are approximated using projection boundary conditions, which involve the stable and unstable manifolds of a steady state solution. The stable manifold of a steady state solution of a delay differential equation (DDE) is infinite-dimensional, a problem which we circumvent by reformulating the end conditions using a special bilinear form. The resulting boundary value problem is solved using a collocation method. We demonstrate results, showing homoclinic orbits in a model for neural activity and travelling wave solutions to the delayed Hodgkin–Huxley equation. Our numerical tests indicate convergence behaviour that corresponds to known theoretical results for ODEs and periodic boundary value problems for DDEs.     

A local projection stabilized method for fictitious domains

In this work a local projection stabilization method is proposed for solving a fictitious domain problem. The method adds a suitable fluctuation term to the formulation, thus yielding the natural space for the Lagrange multiplier stable. Stability and convergence are proved and these results are illustrated with a numerical experiment.     

A domain decomposition method on nested domains and nonmatching grids

A one directionally coupled problem on two nested domains is analyzed. The global domain and the subdomain are discretized by two triangulations that do not match on the subdomain. The connection between the two grids is established by using a stable projection operator onto the interface. An a priori error analysis is carried out and several numerical examples are given. The method is ideally suited for the case of a moving subdomain. © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 20: 374–387, 2004.     

The random projection method for a model problem of combustion with stiff chemical reactions

In this paper we extend the random projection method, recently proposed by the author and S. Jin [J. Comput. Phys. 163 (2000) 216] for under resolved numerical simulations of a qualitative model problem for combustion with stiff chemical reactions:

In this problem, the reaction time is small, making the problem numerically stiff. A classic spurious numerical phenomenon – the incorrect shock speed – occurs when the reaction time scale is not properly resolved numerically. The random projection method is introduced recently to handle this kind of numerical difficulty. The key idea in this method is to randomize the ignition temperature in a suitable domain. Several numerical experiments demonstrate the reliability and robustness of this method.     

On the numerical method for solving a hypersingular integral equation with the computation of the solution gradient

We consider a linear integral equation, which arises when solving the Neumann boundary value problem for the Laplace equation with the representation of the solution in the form of a double layer potential, with a hypersingular integral treated in the sense of Hadamard finite value. We consider the case in which the exterior or interior problem is solved in a domain whose boundary is a closed smooth surface and the integral equation is written over that surface. A numerical scheme for solving the integral equation is constructed with the use of quadrature formulas of the type of the method of discrete singularities with a regularization for the use of an irregular grid. We prove the convergence, uniform over the grid points, of the numerical solutions to the exact solution of the hypersingular equation and, in addition, the uniform convergence of the values of the approximate finite-difference derivative operator on the numerical solution to the values on the projection of the exact solution onto the subspace of grid functions with nodes at the collocation points.     

Two-level pressure projection finite element methods for Navier-Stokes equations with nonlinear slip boundary conditions

The two-level pressure projection stabilized finite element methods for Navier-Stokes equations with nonlinear slip boundary conditions are investigated in this paper, whose variational formulation is the Navier-Stokes type variational inequality problem of the second kind. Based on the P₁-P₁ triangular element and using the pressure projection stabilized finite element method, we solve a small Navier-Stokes type variational inequality problem on the coarse mesh with mesh size H and solve a large Stokes type variational inequality problem for simple iteration or a large Oseen type variational inequality problem for Oseen iteration on the fine mesh with mesh size h. The error analysis obtained in this paper shows that if h=O(H²), the two-level stabilized methods have the same convergence orders as the usual one-level stabilized finite element methods, which is only solving a large Navier-Stokes type variational inequality problem on the fine mesh. Finally, numerical results are given to verify the theoretical analysis.     

Finite element and DG analysis for neutral-type Volterra integro-differential equations with weakly singular kernels

In this paper we analyze the numerical solution of Volterra integro-differential equations of neutral type with weakly singular kernels. We establish a priori error estimations for the finite-element-method semi-discretization of the given problem by defining a suitable Ritz-Volterra projection operator: here, the key point in the proof is the fact that the definition of the Ritz-Volterra projection operator that is not related to the neutral term. We then discuss the discontinuous Galerkin time-stepping method for the semi-discretized equation, together with a fully discretized form.     

Projection formulation of the cavitation problem in elastohydrodynamic lubrication contact

This contribution presents an alternative numerical method on how to find the cavitation region in elastohydrodynamic (EHD) lubrication using an augmented Lagrangian approach. A theoretical framework for the use of a projection formulation instead of a Linear Complementarity Problem (LCP) is given and the application to multibody systems with EHD contacts is shown. With this new formulation the cavitation condition can be covered by an additional algebraic equation. The projection formulation is applied to the steady-state as well as to the dynamic, mass conservative treatment of the cavitation problem. A numerical verification is given for a rigid rotor with unbalance in an elastic bearing. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Analysis and numerical simulation of a nonlinear mathematical model for testing the manoeuvrability capabilities of a submarine

The aim of this work is to provide a mathematical and numerical tool for the analysis of the manoeuvrability capabilities of a submarine. To this end, we consider a suitable optimal control problem with constraints in both state and control variables. The state law is composed of a highly coupled and nonlinear system of twelve ordinary differential equations. Control inputs appear in linear and quadratic form and physically are linked to rudders and propeller forces and moments. We consider a nonlinear Bolza type cost function which represents a commitment between reaching a final desired state and a minimal expense of control. In a first part, following recent ideas in [F. Periago, J. Tiago, A local existence result for an optimal control problem modeling the manoeuvring of an underwater vehicle, Nonlinear Anal. RWA 11 (2010) 2573–2583], we prove a local existence result for the above mentioned optimal control problem. In a second part, we address the numerical resolution of the problem by using a descent method with projection and optimal step-size parameter. To illustrate the performance of the method proposed in this paper and to show its application in a real engineering problem we include three different numerical experiments for a standard manoeuvre.     

Oseen方程的一种新的局部投影稳定化方法

对Oseen方程提出一种新的局部投影稳定化有限元方法,并且速度和压力采用inf-sup稳定的非协调有限元空间逼近．局部投影稳定化项仅加在子网格上(H≥h);与RFB方法相比,该方法稳定性项简单,并且可以克服对流占优．最后,通过实验证明,数值结果和理论结果完全一致．     

A Smooth Regularization of the Projection Formula for Constrained Parabolic Optimal Control Problems

We present a smooth, that is, differentiable regularization of the projection formula that occurs in constrained parabolic optimal control problems. We summarize the optimality conditions in function spaces for unconstrained and control-constrained problems subject to a class of parabolic partial differential equations. The optimality conditions are then given by coupled systems of parabolic PDEs. For constrained problems, a non-smooth projection operator occurs in the optimality conditions. For this projection operator, we present in detail a regularization method based on smoothed sign, minimum and maximum functions. For all three cases, that is, (1) the unconstrained problem, (2) the constrained problem including the projection, and (3) the regularized projection, we verify that the optimality conditions can be equivalently expressed by an elliptic boundary value problem in the space-time domain. For this problem and all three cases we discuss existence and uniqueness issues. Motivated by this elliptic problem, we use a simultaneous space-time discretization for numerical tests. Here, we show how a standard finite element software environment allows to solve the problem and, thus, to verify the applicability of this approach without much implementation effort. We present numerical results for an example problem.     

Navier-Stokes方程的局部投影稳定化方法

将Matthies,Skrzypacz和Tubiska的思想从线性的Oseen方程拓展到了非线性的Navier-Stokes方程,针对不可压缩的定常Navier-Stokes方程,提出了一种局部投影稳定化有限元方法．该方法既克服了对流占优,又绕开了inf-sup条件的限制．给出的局部投影空间既可以定义在两种不同网格上,又可以定义在相同网格上．与其他两级方法相比,定义在同一网格空间上的局部投影稳定化格式更紧凑．在同一网格上,除了给出需要bubble函数来增强的逼近空间外,还特别考虑了两种不需要用bubble函数来增强的新的空间．基于一种特殊的插值技巧,给出了稳定性分析和误差估计．最后,还列举了两个数值算例,进一步验证了理论结果的正确性．     

On the application of alternating projection under the constraint of linear matrix inequalities

In this paper, alternating projection under the constraint oflinear matrix inequalities (LMIs) is investigated to solve thefollowing two problems: finding the intersection of severalconvex LMI sets and designing an output-feedback stabilizingcontroller. Each problem is formulated as a quadratic optimizationproblem under LMI constraints. A numerical algorithm based onthe concept of alternating projection is proposed. The algorithmis demonstrated using a vertical-strip pole-assignment example.     

A self-adaptive projection method for contact problems with the BEM

A self-adaptive algorithm, based on the projection and boundary integral methods, is designed and analyzed for frictionless contact problems in linear elasticity. Using the equivalence between the contact problem and a variational formulation with a projection fixed point problem of infinite dimensions, we develop an iterative algorithm that formulates the contact boundary condition into a sequence of Robin boundary conditions. In order to improve the performance of the method, we propose a self-adaptive rule which updates the penalty parameter automatically. As the iteration process is given by the displacement and the stress on the boundary of the domain, the unknowns of the problem are computed explicitly by using the boundary element method. Both theoretical results and numerical experiments show that the method presented is efficient and robust.     

Analysis of Generalised Galerkin methods in the numerical solution of elliptic equations

The Petrov-Galerkin projection method is outlined for the solution of the linear elliptic equation Lu = f with homogeneous boundary conditions. By choosing appropriate finite dimensional trial and test spaces, the methods of weighted residuals, collocation, and H¹ Galerkin can be interpreted within the Petrov-Galerkin projection method framework. The important question of how best to choose the trial and test functions to suit a particular type of problem is then discussed. Objective criteria associated with the matrix which governs the Petrov-Galerkin numerical process are proposed.     

Projection on Segre varieties and determination of holomorphic mappings between real submanifolds Dedicated to Professor Sheng GONG on the occasion of his 75th birthday

It is shown that a germ of a holomorphic mapping sending a real-analytic generic submanifold of finite type into another is determined by its projection on the Segre variety of the target manifold. A necessary and sufficient condition is given for a germ of a mapping into the Segre variety of the target manifold to be the projection of a holomorphic mapping sending the source manifold into the target. An application to the biholomorphic equivalence problem is also given.