On bounded solutions of transport equation solved with a spectral method

In this article, a spectral method accompanied by finite difference method has been proposed for solving a boundary value problem that accompanies a stationary transport equation. We also prove that the solution is bounded by a value that depends of the source function. The accuracy and computational efficiency of the proposed method are verified with the help of a numerical example. © 2011 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2012     

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.     

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.     

Projected Schur complement method for solving non‐symmetric systems arising from a smooth fictitious domain approach

This paper deals with a fast method for solving large‐scale algebraic saddle‐point systems arising from fictitious domain formulations of elliptic boundary value problems. A new variant of the fictitious domain approach is analyzed. Boundary conditions are enforced by control variables introduced on an auxiliary boundary located outside the original domain. This approach has a significantly higher convergence rate; however, the algebraic systems resulting from finite element discretizations are typically non‐symmetric. The presented method is based on the Schur complement reduction. If the stiffness matrix is singular, the reduced system can be formulated again as another saddle‐point problem. Its modification by orthogonal projectors leads to an equation that can be efficiently solved using a projected Krylov subspace method for non‐symmetric operators. For this purpose, the projected variant of the BiCGSTAB algorithm is derived from the non‐projected one. The behavior of the method is illustrated by examples, in which the BiCGSTAB iterations are accelerated by a multigrid strategy. Copyright © 2007 John Wiley & Sons, Ltd.     

The fictitious domain method with L2‐penalty for the Stokes problem with the Dirichlet boundary condition

We consider the fictitious domain method with L²‐penalty for the Stokes problem with the Dirichlet boundary condition. First, we investigate the error estimates for the penalty method at the continuous level. We obtain the convergence of order in H¹‐norm for the velocity and in L²‐norm for the pressure, where is the penalty parameter. The L²‐norm error estimate for the velocity is upgraded to . Moreover, we derive the a priori estimates depending on for the solution of the penalty problem. Next, we apply the finite element approximation to the penalty problem using the P1/P1 element with stabilization. For the discrete penalty problem, we prove the error estimate in H¹‐norm for the velocity and in L²‐norm for the pressure, where h denotes the discretization parameter. For the velocity in L²‐norm, the convergence rate is improved to . The theoretical results are verified by the numerical experiments.     

Fictitious domain method with penalty for an incompressible fluid

Fictitious domain method shows great advantages when handling problems with complex and constantly varying domains. In this article, we propose an algorithm which extends the fictitious domain method by introducing penalties. Test results with the numerical examples of backward facing step problem and the flow around steady and dynamic cylinder problem show that the algorithm we propose is highly efficient for solving incompressible fluid problems. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2010     

Parallel Galerkin domain decomposition procedures for parabolic equation on general domain

Parallel Galerkin domain decomposition procedures for parabolic equation on general domain are given. These procedures use implicit Galerkin method in the subdomains and simple explicit flux calculation on the interdomain boundaries by integral mean method or extrapolation method to predict the inner‐boundary conditions. Thus, the parallelism can be achieved by these procedures. These procedures are conservative both in the subdomains and across interboundaries. The explicit nature of the flux prediction induces a time‐step limitation that is necessary to preserve stability, but this constraint is less severe than that for a fully explicit method. L²‐norm error estimates are derived for these procedures. Compared with the work of Dawson and Dupont [Math Comp 58 (1992), 21–35], these L²‐norm error estimates avoid the loss of H^?1/2 factor. Experimental results are presented to confirm the theoretical results. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009     

On the homotopy analysis method for solving a particle transport equation

The purpose of this paper consists in the finding of the solution for a stationary neutron transport equation that is accompanied by the homogeneous boundary conditions, using the techniques of homotopy analysis method (HAM) and a numerical integration formula. Also, algorithm presented can be used for solving the integral–differential equations in which the unknown function depends on two variables, such as a radiative transfer equation. Results of a numerical example illustrate the accuracy and computational efficiency of the new proposed method.     

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.     

He's variational iteration method for the solution of nonlinear Newell-Whitehead-Segel equation

In this paper, we apply He''s Variational iteration method (VIM) for solving nonlinear Newell-Whitehead-Segel equation. By using this method three different cases of Newell-Whitehead-Segel equation have been discussed. Comparison of the obtained result with exact solutions shows that the method used is an effective and highly promising method for solving different cases of nonlinear Newell-Whitehead-Segel equation.     

Conservative‐dissipative approximation schemes for a generalized Kramers equation

We propose three new discrete variational schemes that capture the conservative‐dissipative structure of a generalized Kramers equation. The first two schemes are single‐step minimization schemes, whereas the third one combines a streaming and a minimization step. The cost functionals in the schemes are inspired by the rate functional in the Freidlin‐Wentzell theory of large deviations for the underlying stochastic system. We prove that all three schemes converge to the solution of the generalized Kramers equation. Copyright © 2013 John Wiley & Sons, Ltd.     

An approximation to the solution of telegraph equation by variational iteration method

The variational iteration method (VIM) has been applied to solve many functional equations. In this article, this method is applied to obtain an approximate solution for the Telegraph equation. Some examples are presented to show the ability of the proposed method. The results of applying VIM are exactly the same as those obtained by Adomian decomposition method. It seems less computation is needed in proposed method.© 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009     

On a fictitious domain method with distributed Lagrange multiplier for interface problems

In this paper we propose a new variational formulation for an elliptic interface problem and discuss its finite element approximation. Our formulation fits within the framework of fictitious domain methods with distributed Lagrange multipliers. For the underlying mixed scheme we prove stability and convergence. Some preliminary numerical tests confirm the theoretical investigations.     

A homotopy perturbation method for solving a neutron transport equation

The purpose of this paper consists in the finding of the solution for a stationary transport equation using the techniques of homotopy perturbation method (HPM). The results of a numerical example illustrate the accuracy and computational efficiency of the new proposed method.     

Stochastic process model for solute transport and the associated transport equation

A mathematical model of Lagrangian motions of a particle in turbulent flows is developed on the basis of a stochastic differential equation. The model expresses uncertainties involved in turbulence by standard Brownian motion. Because the model does not guarantee smoothness of the path of the particle, local velocity is newly defined so as to be suitable for observation of a velocity time series at a fixed point. Then, it is shown that the newly defined local velocity is governed by a Gaussian distribution. In addition, an estimation method of the turbulent diffusion coefficient involved in the model is proposed by using the local velocity. The estimation method does not require tracer experiments. In order to assess the validity of the proposed local velocity, velocity measurements with three-dimensional acoustic Doppler velocimeters were conducted in agricultural drainage canals. Also, the turbulent diffusion coefficient was estimated by the derived time series of the observed local velocity. Finally, a transport equation of conservative solute is derived by using the linearity of the Kolmogorov forward equation without using gradient-type lows.     

On a thermoelastic plate equation in an exterior domain

Obtained are the existence of solutions and the local energy decay of a linear thermoelastic plate equation in a 3 dim. exterior domain. The thermoplate equation is formulated as a Sobolev equation in the abstract framework. Our proof of the existence theorem is based on an argument due to Goldstein (Semigroups of Linear Operators and Applications. Oxford University Press: New York, 1985). To obtain the local energy decay, we use the commutation method in order to treat the high‐frequency part and a precise expansion of the resolvent operator obtained by constructing the parametrix in order to treat the low‐frequency. Copyright © 2002 John Wiley & Sons, Ltd.     

On a parabolic equation in a triangular domain

We prove the optimal regularity, in Sobolev spaces, of the solution of a parabolic equation set in a triangular domain T. The right-hand term of the equation is taken in Lebesgue space L^p(T). The method of operators sums in the non-commutative case is referred to.     

Analysis of segregated boundary‐domain integral equations for variable‐coefficient problems with cracks

Segregated direct boundary‐domain integral equation (BDIE) systems associated with mixed, Dirichlet and Neumann boundary value problems (BVPs) for a scalar “Laplace” PDE with variable coefficient are formulated and analyzed for domains with interior cuts (cracks). The main results established in the paper are the BDIE equivalence to the original BVPs and invertibility of the BDIE operators in the corresponding Sobolev spaces. © 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2010     

Differential quadrature domain decomposition method for problems on a triangular domain

The differential quadrature method (DQM) has been studied for years and it has been shown by many researchers that the DQM is an attractive numerical method with high efficiency and accuracy. The conventional DQM is mostly effective for one‐dimensional and multidimensional problems with geometrically regular domains. But to deal with problems on a triangular domain, we will meet difficulties. In this article we will study how to solve problems on a triangular domain by using DQM combined with the domain decomposition method (DDM). © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2005     

非定常BOLTZMANN方程的谱有限元方法

In this paper,a new numerical method,the coupling method of spherical harmonic function spectral and finite elements,for a unsteady transport equation is dlscussed,and the error analysis of this scheme is proved.