Legendre spectral method for solving mixed boundary value problems on rectangle

In this paper, we develop a spectral method for mixed inhomogeneous Dirichlet/Neumann/Robin boundary value problems defined on rectangle. Some results on two‐dimensional Legendre approximation in Jacobi‐weighted Sobolev space are established. As examples of applications, spectral schemes are provided for two model problems with mixed inhomogeneous boundary conditions. The spectral accuracy in space of proposed algorithms are proved. Efficient implementations are presented. Numerical results demonstrate their high accuracy and confirm the theoretical analysis well. Copyright © 2016 John Wiley & Sons, Ltd.     

A uniformly convergent hybrid scheme for singularly perturbed system of reaction-diffusion Robin type boundary-value problems

This paper deals with the study on system of reaction diffusion differential equations for Robin or mixed type boundary value problems (MBVPs). A cubic spline approximation has been used to obtain the difference scheme for the system of MBVPs, on a piecewise uniform Shishkin mesh defined in the whole domain. It has been shown that our proposed scheme, i.e., central difference approximation for outer region with cubic spline approximation for inner region of boundary layers, leads to almost second order parameter uniform convergence whereas the standard method i.e., the forward-backward approximation for mixed boundary conditions with central difference approximation inside the domain leads to almost first order convergence on Shishkin mesh. Numerical results are provided to show the efficiency and accuracy of these methods.     

Initial‐Boundary Value Problems for the Defocusing Nonlinear Schrödinger Equation in the Semiclassical Limit

Initial‐boundary value problems for integrable nonlinear partial differential equations have become tractable in recent years due to the development of so‐called unified transform techniques. The main obstruction to applying these methods in practice is that calculation of the spectral transforms of the initial and boundary data requires knowledge of too many boundary conditions, more than are required to make the problem well‐posed. The elimination of the unknown boundary values is frequently addressed in the spectral domain via the so‐called global relation, and types of boundary conditions for which the global relation can be solved are called linearizable. For the defocusing nonlinear Schrödinger equation, the global relation is only known to be explicitly solvable in rather restrictive situations, namely homogeneous boundary conditions of Dirichlet, Neumann, and Robin (mixed) type. General nonhomogeneous boundary conditions are not known to be linearizable. In this paper, we propose an explicit approximation for the nonlinear Dirichlet‐to‐Neumann map supplied by the defocusing nonlinear Schrödinger equation and use it to provide approximate solutions of general nonhomogeneous boundary value problems for this equation posed as an initial‐boundary value problem on the half‐line. Our method sidesteps entirely the solution of the global relation. The accuracy of our method is proven in the semiclassical limit, and we provide explicit asymptotics for the solution in the interior of the quarter‐plane space‐time domain.     

Accurate spectral solutions for the parabolic and elliptic partial differential equations by the ultraspherical tau method

We present a double ultraspherical spectral methods that allow the efficient approximate solution for the parabolic partial differential equations in a square subject to the most general inhomogeneous mixed boundary conditions. The differential equations with their boundary and initial conditions are reduced to systems of ordinary differential equations for the time-dependent expansion coefficients. These systems are greatly simplified by using tensor matrix algebra, and are solved by using the step-by-step method. Numerical applications of how to use these methods are described. Numerical results obtained compare favorably with those of the analytical solutions. Accurate double ultraspherical spectral approximations for Poisson's and Helmholtz's equations are also noted. Numerical experiments show that spectral approximation based on Chebyshev polynomials of the first kind is not always better than others based on ultraspherical polynomials.     

Some progress in spectral methods

In this paper,we review some results on the spectral methods.We frst consider the Jacobi spectral method and the generalized Jacobi spectral method for various problems,including degenerated and singular diferential equations.Then we present the generalized Jacobi quasi-orthogonal approximation and its applications to the spectral element methods for high order problems with mixed inhomogeneous boundary conditions.We also discuss the related spectral methods for non-rectangular domains and the irrational spectral methods for unbounded domains.Next,we consider the Hermite spectral method and the generalized Hermite spectral method with their applications.Finally,we consider the Laguerre spectral method and the generalized Laguerre spectral method for many problems defned on unbounded domains.We also present the generalized Laguerre quasi-orthogonal approximation and its applications to certain problems of non-standard type and exterior problems.     

Composite Laguerre-Legendre spectral method for exterior problems

In this paper, we propose a composite Laguerre-Legendre spectral method for two-dimensional exterior problems. Results on the composite Laguerre-Legendre approximation, which is a set of piecewise mixed approximations coupled with domain decomposition, are established. These results play important roles in the related spectral methods for exterior problems. As examples of applications, the composite spectral schemes are provided for two model problems, with the convergence analysis. An efficient implementation is described. Numerical results demonstrate the spectral accuracy in space of this new approach, and confirm the analysis. The approximation results and techniques developed in this paper are also applicable to other problems defined on unbounded domains.     

Solving 2D time‐fractional diffusion equations by a pseudospectral method and Mittag‐Leffler function evaluation

Two‐dimensional time‐fractional diffusion equations with given initial condition and homogeneous Dirichlet boundary conditions in a bounded domain are considered. A semidiscrete approximation scheme based on the pseudospectral method to the time‐fractional diffusion equation leads to a system of ordinary fractional differential equations. To preserve the high accuracy of the spectral approximation, an approach based on the evaluation of the Mittag‐Leffler function on matrix arguments is used for the integration along the time variable. Some examples along with numerical experiments illustrate the effectiveness of the proposed approach. Copyright © 2016 John Wiley & Sons, Ltd.     

Thermally coupled,stationary, incompressible MHD flow; existence,uniqueness, and finite element approximation

This article addresses the questions of existence, uniqueness, and finite element approximation (including some computational aspects) of solutions to the equations of steady-state magnetohy-drodynamic (MHD) when buoyancy effects due to temperature differences in the flow cannot be neglected. We couple the MHD equations to the heat equation and employ the well-known Boussinesq approximation. We consider the equations posed on a bounded three-dimensional domain. The boundary conditions for the velocity are of Dirichlet type; the boundary conditions for the temperature are mixed (of Dirichlet type and of Neumann type); we also specify the normal component of the magnetic field and tangential component of the electric field on the boundary. We point out that these problems are relevant to many physical phenomena such as the cooling of nuclear reactors by electrically conducting fluids, continuous metal casting, crystal growth, and semi-conductor manufacture. © 1995 John Wiley & Sons, Inc.     

Stability estimates for h-p spectral element methods for general elliptic problems on curvilinear domains

In this paper we show that the h-p spectral element method developed in [3,8,9] applies to elliptic problems in curvilinear polygons with mixed Neumann and Dirichlet boundary conditions provided that the Babuska-Brezzi inf-sup conditions are satisfied. We establish basic stability estimates for a non-conforming h-p spectral element method which allows for simultaneous mesh refinement and variable polynomial degree. The spectral element functions are non-conforming if the boundary conditions are Dirichlet. For problems with mixed boundary conditions they are continuous only at the vertices of the elements. We obtain a stability estimate when the spectral element functions vanish at the vertices of the elements, which is needed for parallelizing the numerical scheme. Finally, we indicate how the mesh refinement strategy and choice of polynomial degree depends on the regularity of the coefficients of the differential operator, smoothness of the sides of the polygon and the regularity of the data to obtain the maximum accuracy achievable. An erratum to this article is available at .     

Regularity and a priori error analysis of a Ventcel problem in polyhedral domains

We consider the regularity of a mixed boundary value problem for the Laplace operator on a polyhedral domain, where Ventcel boundary conditions are imposed on one face of the polyhedron and Dirichlet boundary conditions are imposed on the complement of that face in the boundary. We establish improved regularity estimates for the trace of the variational solution on the Ventcel face and use them to derive a decomposition of the solution into a regular and a singular part that belongs to suitable weighted Sobolev spaces. This decomposition, in turn, via interpolation estimates both in the interior as well as on the Ventcel face, allows us to perform an a priori error analysis for the finite element approximation of the solution on anisotropic graded meshes. Numerical tests support the theoretical analysis. Copyright © 2016 John Wiley & Sons, Ltd.     

Semi-analytical hybrid approach for the simulation of layered waveguide with a partially debonded piezoelectric structure

This paper presents an efficient semi-analytical hybrid approach for simulating the dynamic interaction of perfectly bonded or damaged piezoelectric structures with a layered elastic waveguide. In the proposed approach, the frequency domain spectral element method is utilized for the discretization of the finite-sized surface mounted piezoelectric structure, and the semi-analytical boundary integral method is employed for the evaluation of wave phenomena in the host laminate structure. While the spectral element method allows cost-effective simulation of dynamics of a complex-shaped transducer (e.g. curvilinear or with wrapped electrodes), the analytically-based technique reliably describes wave excitation and propagation in multi-layered structures. The coupling of these methods is achieved through the rigorous fulfillment of the boundary conditions at the area of waveguide-transducer contact. Three various combinations of approximation polynomials and surface-load interpolation functions are applied in order to obtain the solution in a frequency domain. The time-domain solution is evaluated employing the inverse Laplace transform. Convergence of the method is confirmed for different bonding conditions. The paper demonstrates the efficiency of the proposed method for the multi-parameter analysis of the dependence of the resonance characteristics on the debonding parameters and contact conditions. The approach can be used for such a crucial task as diagnosing failures of piezoelectric devices incorporated into a structural health monitoring system based on guided waves.     

Spectral stability results for higher-order operators under perturbations of the domain

We analyze the spectral behavior of higher-order elliptic operators when the domain is perturbed. We provide general spectral stability results for Dirichlet and Neumann boundary conditions. Moreover, we study the bi-harmonic operator with the so-called intermediate boundary conditions. We give special attention to this last case and analyze its behavior when the boundary of the domain has some oscillatory behavior. We will show that there is a critical oscillatory behavior and that the limit problem depends on whether we are above, below or just sitting on this critical value.     

A multidomain spectral approximation of elliptic equations

A spectral approximation for the Poisson equation defined on Ω = ]?1, 1[×] ?1,1[ is studied. The domain Ω is decomposed into two rectangular regions and the equation is collocated at the Legendre nodes in each domain. On the common boundary of the two subdomains, suitable conditions are imposed in order to obtain a unique solution from the resulting linear system. Different values of the discretization parameters are allowed in each rectangle. We prove the stability of the scheme and give convergence estimates. The rate of convergence in a single subdomain, depends only on the regularity of the exact solution therein. An efficient preconditioning matrix is proposed.     

Finite element approximation of spectral problems with Neumann boundary conditions on curved domains

This paper deals with the finite element approximation of the spectral problem for the Laplace equation with Neumann boundary conditions on a curved nonconvex domain . Convergence and optimal order error estimates are proved for standard piecewise linear continuous elements on a discrete polygonal domain in the framework of the abstract spectral approximation theory.

     

Asymptotics of the solution of Laplace’s equation with third-type boundary conditions on the boundaries of two small holes

A boundary value problem for Laplace’s equation in a bounded domain with two small holes is considered. Third-type boundary conditions are set on the boundaries of the holes. A Neumann condition is specified on the outer boundary of the domain. A uniform asymptotic approximation of the solution is constructed and justified up to an arbitrary power of a small parameter.     

A posteriori error analysis for elliptic pdes on domains with complicated structures

Summary. The discretisation of boundary value problems on complicated domains cannot resolve all geometric details such as small holes or pores. The model problem of this paper consists of a triangulated polygonal domain with holes of a size of the mesh-width at most and mixed boundary conditions for the Poisson equation. Reliable and efficient a posteriori error estimates are presented for a fully numerical discretisation with conforming piecewise affine finite elements. Emphasis is on technical difficulties with the numerical approximation of the domain and their influence on the constants in the reliability and efficiency estimates. Mathematics Subject Classification (2000):65N30, 65R20, 73C50Received: 28, June 2001     

Artificial boundary conditions for elliptic systems on polyhedral truncation surfaces

For a sufficiently broad class of formally selfadjoint boundary value problems in the domains with conical outlets to infinity, including exterior boundary value problems, we suggest an algorithm for constructing some artificial boundary conditions on polyhedral truncation surfaces that guarantees higher precision of approximation as the infinite domain is replaced with a large but bounded domain. The errors are estimated. Anisotropic 3-dimensional elasticity problems are discussed as an example.     

Discontinuous Galerkin method for nonlinear convection-diffusion problems with mixed Dirichlet-Neumann boundary conditions

The paper is devoted to the analysis of the discontinuous Galerkin finite element method (DGFEM) applied to the space semidiscretization of a nonlinear nonstationary convection-diffusion problem with mixed Dirichlet-Neumann boundary conditions. General nonconforming meshes are used and the NIPG, IIPG and SIPG versions of the discretization of diffusion terms are considered. The main attention is paid to the impact of the Neumann boundary condition prescribed on a part of the boundary on the truncation error in the approximation of the nonlinear convective terms. The estimate of this error allows to analyse the error estimate of the method. The results obtained represent the completion and extension of the analysis from V. Dolej?í, M. Feistauer, Numer. Funct. Anal. Optim. 26 (2005), 349–383, where the truncation error in the approximation of the nonlinear convection terms was proved only in the case when the Dirichlet boundary condition on the whole boundary of the computational domain was considered.     

Error bounds for the finite element approximation of an incompressible material in an unbounded domain

Summary. In this paper we design high-order local artificial boundary conditions and present error bounds for the finite element approximation of an incompressible elastic material in an unbounded domain. The finite element approximation is formulated in a bounded computational domain using a nonlocal approximate artificial boundary condition or a local one. In fact there are a family of nonlocal approximate artificial boundary conditions with increasing accuracy (and computational cost) and a family of local ones for a given artificial boundary. Our error bounds indicate how the errors of the finite element approximations depend on the mesh size, the terms used in the approximate artificial boundary condition and the location of the artificial boundary. Numerical examples of an incompressible elastic material outside a circle in the plane is presented. Numerical results demonstrate the performance of our error bounds. Received August 31, 1998 / Revised version received November 6, 2001 / Published online March 8, 2002     

THE APPROXIMATIONS OF THE EXACT BOUNDARY CONDITION AT AN ARTIFICIAL BOUNDARY FOR LINEARIZED INCOMPRESSIBLE VISCOUS FLOWS

1.IntroductionManyproblemsarisinginfluidmechanicsaregiveninanunboundeddomain,suchasfluidflowaroundobstacles.Whencomputingthenumericalsolutionsoftheseproblems,oneoftenintroducesartificialboundariesandsetsupaxtificialboundaryconditionsonthem.Thentheoriginal…