-curved finite elements and applications to plate and shell problems

Many thin-plate and thin-shell problems are set on plane reference domains with a curved boundary. Their approximation by conforming finite-elements methods requires ¹-curved finite elements entirely compatible with the associated ¹-rectilinear finite elements. In this contribution we introduce a ¹-curved finite element compatible with the P₅-Argyris element, we study its approximation properties, and then, we use such an element to approximate the solution of thin-plate or thin-shell problems set on a plane-curved boundary domain. We prove the convergence and we get a priori asymptotic error estimates which show the very high degree of accuracy of the method. Moreover we obtain criteria to observe when choosing the numerical integration schemes in order to preserve the order of the error estimates obtained for exact integration.     

A high order uniformly convergent alternating direction scheme for time dependent reaction–diffusion singularly perturbed problems

In this work we design and analyze an efficient numerical method to solve two dimensional initial-boundary value reaction–diffusion problems, for which the diffusion parameter can be very small with respect to the reaction term. The method is defined by combining the Peaceman and Rachford alternating direction method to discretize in time, together with a HODIE finite difference scheme constructed on a tailored mesh. We prove that the resulting scheme is ε-uniformly convergent of second order in time and of third order in spatial variables. Some numerical examples illustrate the efficiency of the method and the orders of uniform convergence proved theoretically. We also show that it is easy to avoid the well-known order reduction phenomenon, which is usually produced in the time integration process when the boundary conditions are time dependent. This research has been partially supported by the project MEC/FEDER MTM2004-01905 and the Diputación General de Aragón.     

An iterative method based on equation decomposition for the fourth‐order singular perturbation problem

In this article, we propose an iterative method based on the equation decomposition technique ( 1 ) for the numerical solution of a singular perturbation problem of fourth‐order elliptic equation. At each step of the given method, we only need to solve a boundary value problem of second‐order elliptic equation and a second‐order singular perturbation problem. We prove that our approximate solution converges to the exact solution when the domain is a disc. Our numerical examples show the efficiency and accuracy of our method. Our iterative method works very well for singular perturbation problems, that is, the case of 0 < ε ? 1, and the convergence rate is very fast. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013     

Asymptotic analysis for target asset portfolio allocation with small transaction costs

In this paper we discuss the asset allocation in the presence of small proportional transaction costs. The objective is to keep the asset portfolio close to a target portfolio and at the same time to reduce the trading cost in doing so. We derive the variational inequality and prove a verification theorem. Furthermore, we apply the second order asymptotic expansion method to characterize explicitly the optimal no transaction region when the transaction cost is small and show that the boundary points are asymmetric in relation to the target portfolio position, in contrast to the symmetric relation when only the first order asymptotic expansion method is used, and the leading order is a constant proportion of the cubic root of the small transaction cost. In addition, we use the asymptotic results for the boundary points and obtain an expansion for the value function. The results are illustrated in the numerical example.     

Crouzeix–Raviart boundary elements

This paper establishes a foundation of non-conforming boundary elements. We present a discrete weak formulation of hypersingular integral operator equations that uses Crouzeix–Raviart elements for the approximation. The cases of closed and open polyhedral surfaces are dealt with. We prove that, for shape regular elements, this non-conforming boundary element method converges and that the usual convergence rates of conforming elements are achieved. Key ingredient of the analysis is a discrete Poincaré–Friedrichs inequality in fractional order Sobolev spaces. A numerical experiment confirms the predicted convergence of Crouzeix–Raviart boundary elements. Norbert Heuer is supported by Fondecyt-Chile under grant no. 1080044. F.-J. Sayas is partially supported by MEC-FEDER Project MTM2007-63204 and Gobierno de Aragón (Grupo Consolidado PDIE).     

Finite Difference Method for Reaction-Diffusion Nonlocal Boundary Conditions Equation with

In this paper, we present a numerical approach to a class of nonlinear reactiondiffusion equations with nonlocal Robin type boundary conditions by finite difference methods. A second-order accurate difference scheme is derived by the method of reduction of order. Moreover, we prove that the scheme is uniquely solvable and convergent with the convergence rate of order two in a discrete L2-norm. A simple numerical example is given to illustrate the efficiency of the proposed method.     

On spline collocation methods for boundary integral equations in the plane

Nowadays boundary elemen; methods belong to the most popular numerical methods for solving elliptic boundary value problems. They consist in the reduction of the problem to equivalent integral equations (or certain generalizations) on the boundary Γ of the given domain and the approximate solution of these boundary equations. For the numerical treatment the boundary surface is decomposed into a finite number of segments and the unknown functions are approximated by corresponding finite elements and usually determined by collocation and Galerkin procedures. One finds the least difficulties in the theoretical foundation of the convergence of Galerkin methods for certain classes of equations, whereas the convergence of collocation methods, which are mostly used in numerical computations, has yet been proved only for special equations and methods. In the present paper we analyse spline collocation methods on uniform meshes with variable collocation points for one-dimensional pseudodifferential equations on a closed curve with convolutional principal parts, which encompass many classes of boundary integral equations in the plane. We give necessary and sufficient conditions for convergence and prove asymptotic error estimates. In particular we generalize some results on nodal and midpoint collocation obtained in [2], [7] and [8]. The paper is organized as follows. In Section 1 we formulate the problems and the results, Section 2 deals with spline interpolation in periodic Sobolev spaces, and in Section 3 we prove the convergence theorems for the considered collocation methods.     

Finite Difference Method for Reaction-Diffusion Equation with Nonlocal Boundary Conditions

In this paper, we present a numerical approach to a class of nonlinear reaction-diffusion equations with nonlocal Robin type boundary conditions by finite difference methods. A second-order accurate difference scheme is derived by the method of reduction of order. Moreover, we prove that the scheme is uniquely solvable and convergent with the convergence rate of order two in a discrete L2-norm. A simple numerical example is given to illustrate the efficiency of the proposed method.     

A fractional step method for 2D parabolic convection-diffusion singularly perturbed problems: uniform convergence and order reduction

In this work, we are concerned with the efficient resolution of two dimensional parabolic singularly perturbed problems of convection-diffusion type. The numerical method combines the fractional implicit Euler method to discretize in time on a uniform mesh and the classical upwind finite difference scheme, defined on a Shishkin mesh, to discretize in space. We consider general time-dependent Dirichlet boundary conditions, and we show that classical evaluations of the boundary conditions cause an order reduction in the consistency of the time integrator. An appropriate correction for the evaluations of the boundary data permits to remove such order reduction. Using this correction, we prove that the fully discrete scheme is uniformly convergent of first order in time and of almost first order in space. Some numerical experiments, which corroborate in practice the robustness and the efficiency of the proposed numerical algorithm, are shown; from them, we bring to light the influence in practice of the two options for the boundary data considered here, which is in agreement with the theoretical results.     

曲边区域上的多边形网格间断有限元离散及其多重网格算法

本文利用多边形网格上的间断有限元方法离散二阶椭圆方程,在曲边区域上,采用多条直短边逼近曲边的以直代曲的策略,实现了高阶元在能量范数下的最优收敛.本文还将这一方法用于带曲边界面问题的求解,同样得到高阶元的最优收敛.此外我们还设计并分析了这一方法的\linebreakW-cycle和Variable V-cycle多重网格预条件方法,证明当光滑次数足够多时,多重网格预条件算法一致收敛.最后给出了数值算例,证实该算法的可行性并验证了理论分析的结果.     

A numerical approach for the Poisson equation in a planar domain with a small inclusion

We consider the Poisson equation in a domain with a small inclusion. We present a simple numerical method, based on asymptotic analysis, which allows to approximate robustly the far field of the solution as the size of the inclusion goes to zero without any mesh adaptation procedure. The discretization is based on a fully standard Galerkin approach such as finite elements. We prove stability and consistency of the numerical method and provide error estimates. We end the paper with numerical experiments illustrating the efficiency of the technique.     

Augmented Galerkin schemes for the numerical solution of scattering by small obstacles

We are interested in the problem of a bidimensional acoustic wave propagation in a medium including a small obstacle with homogeneous Dirichlet boundary condition. We present and analyse a numerical scheme suitable for finite elements that does not suffer from numerical locking, and takes the presence of the small obstacle into account. It is based on the fictitious domain method combined with matched asymptotic expansions.     

Analysis of a meshless method for the time fractional diffusion-wave equation

In this paper a numerical technique is proposed for solving the time fractional diffusion-wave equation. We obtain a time discrete scheme based on finite difference formula. Then, we prove that the time discrete scheme is unconditionally stable and convergent using the energy method and the convergence order of the time discrete scheme is \(\mathcal {O}(\tau ^{3-\alpha })\). Firstly, we change the main problem based on Dirichlet boundary condition to a new problem based on Robin boundary condition and then, we consider a semi-discrete scheme with Robin boundary condition and show when \(\beta \rightarrow +\infty \) solution of the main semi-discrete problem with Dirichlet boundary condition is convergent to the solution of the new semi-discrete problem with Robin boundary condition. We consider the new semi-discrete problem with Robin boundary condition and use the meshless Galerkin method to approximate the spatial derivatives. Finally, we obtain an error bound for the new problem. We prove that convergence order of the numerical scheme based on Galekin meshless is \(\mathcal {O}(h)\). In the considered method the appeared integrals are approximated using Gauss Legendre quadrature formula. The main aim of the current paper is to obtain an error estimate for the meshless Galerkin method based on the radial basis functions. Numerical examples confirm the efficiency and accuracy of the proposed scheme.     

Runge-Kutta methods without order reduction for linear initial boundary value problems

Summary. It is well-known the loss of accuracy when a Runge–Kutta method is used together with the method of lines for the full discretization of an initial boundary value problem. We show that this phenomenon, called order reduction, is caused by wrong boundary values in intermediate stages. With a right choice, the order reduction can be avoided and the optimal order of convergence in time is achieved. We prove this fact for time discretizations of abstract initial boundary value problems based on implicit Runge–Kutta methods. Moreover, we apply these results to the full discretization of parabolic problems by means of Galerkin finite element techniques. We present some numerical examples in order to confirm that the optimal order is actually achieved. Received July 10, 2000 / Revised version received March 13, 2001 / Published online October 17, 2001     

Efficient treatment of stationary free boundary problems

In the present paper we consider the efficient treatment of free boundary problems by shape optimization. We reformulate the free boundary problem as shape optimization problem. A second order shape calculus enables us to analyze the shape problem under consideration and to prove convergence of a Ritz–Galerkin approximation of the shape. We show that Newton's method requires only access to the underlying state function on the boundary of the domain. We compute these data by boundary integral equations which are numerically solved by a fast wavelet Galerkin scheme. Numerical results prove that we succeeded in finding a fast and robust algorithm for solving the considered class of problems.     

Error estimates for a discretized Galerkin method for a boundary integral equation in two dimensions

We present a priori and a posteriori estimates for the error between the Galerkin and a discretized Galerkin method for the boundary integral equation for the single layer potential on the square plate. Using piecewise constant finite elements on a rectangular mesh we study the error coming from numerical integration. The crucial point of our analysis is the estimation of some error constants, and we demonstrate that this is necessary if our methods are to be used. After the determination of these constants we are in the position to prove invertibility and quasioptimal convergence results for our numerical scheme, if the chosen numerical integration formulas are sufficiently precise. © 1992 John Wiley & Sons, Inc.     

A hybrid high-order formulation for a Neumann problem on polytopal meshes

In this work, we study a hybrid high-order (HHO) method for an elliptic diffusion problem with Neumann boundary condition. The proposed method has several features, such as: (a) the support of arbitrary approximation order polynomial at mesh elements and faces on polytopal meshes, (b) the design of a local (element-wise) potential reconstruction operator and a local stabilization term, that weakly enforces the matching between local element- and face-based on degrees of freedom, and (c) cheap computational cost, thanks to static condensation and compact stencil. We prove the well-posedness of our HHO formulation, and obtain the optimal error estimates, according to previous study. Implementation aspects are thoroughly discussed. Finally, some numerical examples are provided, which are in agreement with our theoretical results.     

Discontinuous Least-Squares finite element method for the Div-Curl problem

In this paper, we consider the div-curl problem posed on nonconvex polyhedral domains. We propose a least-squares method based on discontinuous elements with normal and tangential continuity across interior faces, as well as boundary conditions, weakly enforced through a properly designed least-squares functional. Discontinuous elements make it possible to take advantage of regularity of given data (divergence and curl of the solution) and obtain convergence also on nonconvex domains. In general, this is not possible in the least-squares method with standard continuous elements. We show that our method is stable, derive a priori error estimates, and present numerical examples illustrating the method.     

The convergence of a direct BEM for the plane mixed boundary value problem of the Laplacian

Summary In this paper the convergence analysis of a direct boundary elecment method for the mixed boundary value problem for Laplace equation in a smooth plane domain is given. The method under consideration is based on the collocation solution by constant elements of the corresponding system of boundary integral equations. We prove the convergence of this method, provide asymptotic error estimates for the BEM-solution and give some numerical examples.     

Finite element analysis of the vibration problem of a plate coupled with a fluid

Summary. We consider the approximation of the vibration modes of an elastic plate in contact with a compressible fluid. The plate is modelled by Reissner-Mindlin equations while the fluid is described in terms of displacement variables. This formulation leads to a symmetric eigenvalue problem. Reissner-Mindlin equations are discretized by a mixed method, the equations for the fluid with Raviart-Thomas elements and a non conforming coupling is used on the interface. In order to prove that the method is locking free we consider a family of problems, one for each thickness , and introduce appropriate scalings for the physical parameters so that these problems attain a limit when . We prove that spurious eigenvalues do not arise with this discretization and we obtain optimal order error estimates for the eigenvalues and eigenvectors valid uniformly on the thickness parameter t. Finally we present numerical results confirming the good performance of the method. Received February 4, 1998 / Revised version received May 26, 1999 / Published online June 21, 2000