Construction of locally conservative fluxes for the SUPG method

We consider the construction of locally conservative fluxes by means of a simple postprocessing technique obtained from the finite element solutions of advection diffusion equations. It is known that a naive calculation of fluxes from these solutions yields nonconservative fluxes. We consider two finite element methods: the usual continuous Galerkin finite element method for solving nondominating advection diffusion equations and the streamline upwind/Petrov‐Galerkin method for solving advection dominated problems. We then describe the postprocessing technique for constructing conservative fluxes from the numerical solutions of the general variational formulation. The postprocessing technique requires solving an auxiliary Neumann boundary value problem on each element independently and it produces a locally conservative flux on a vertex centered dual mesh relative to the finite element mesh. We provide a convergence analysis for the postprocessing technique. Performance of the technique and the convergence behavior are demonstrated through numerical examples including a set of test problems for advection diffusion equations, advection dominated equations, and drift‐diffusion equations. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 1971–1994, 2015     

Solvability of the clamped Euler–Bernoulli beam equation

In this study, solvability of the initial boundary value problem for general form Euler–Bernoulli beam equation which includes also moving point-loads is investigated. The complete proof of an existence and uniqueness properties of the weak solution of the considered equation with Dirichlet type boundary conditions is derived. The method used here is based on Galerkin approximation which is the main tool for the weak solution theory of linear evolution equations as well as in derivation of a priori estimate for the approximate solutions. All steps of the proposed technique are explained in detail.     

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.     

Boundary element methods for potential problems with nonlinear boundary conditions

Galerkin boundary element methods for the solution of novel first kind Steklov-Poincaré and hypersingular operator boundary integral equations with nonlinear perturbations are investigated to solve potential type problems in two- and three-dimensional Lipschitz domains with nonlinear boundary conditions. For the numerical solution of the resulting Newton iterate linear boundary integral equations, we propose practical variants of the Galerkin scheme and give corresponding error estimates. We also discuss the actual implementation process with suitable preconditioners and propose an optimal hybrid solution strategy.

     

A boundary integral equation method for three-dimensional crack problems in elasticity

This paper presents a solution procedure for three-dimensional crack problems via first kind boundary integral equations on the crack surface. The Dirichlet (Neumann) problem is reduced to a system of integral equations for the jump of the traction (of the field) across the crack surface. The calculus of pseudodifferential operators is used to derive existence and regularity of the solutions of the integral equations. With the concept of the principal symbol and the Wiener-Hopf technique we derive the explicit behavior of the densities of the integral equations near the edge of the crack surface. Based on the detailed regularity results we show how to improve the boundary element Galerkin method for our integral equations. Quasi-optimal asymptotic estimates for the Galerkin error are given.     

二维N－S方程的Fourier非线性Galerkin方法

本文对周期边界条件Ｎａｖｉｅｒ－Ｓｔｏｋｅｓ方程，证明了其Ｆｏｕｒｉｅｒ非线性Ｇａｌｅｒｋｉｎ逼近解的存在唯一性，同时给出了逼近解的误差估计．     

A stabilized semi-implicit Galerkin scheme for Navier-Stokes equations

By introducing a time relaxation term for the time derivative of higher frequency components, we proposed a stabilized semi-implicit Galerkin scheme for evolutionary Navier-Stokes equations in this paper. Analysis shows that such a scheme has weaker stability conditions than that of a classical semi-implicit Galerkin scheme and, when a suitable relaxation parameter σ is chosen, it generates an approximate solution with the same accuracy as the classical one. That means the proposed scheme might use a larger time step to generate a bounded approximate solution. Thus it is more suitable for long time simulations.     

An Element-Free Galerkin-Based Multi-objective Optimization of Laminated Composite Plates

An element-free Galerkin method is presented to analyze isotropic and laminated composite plates. This method employs the moving least square technique to approximate functions. In the analysis procedure, a collocation method is used to enforce boundary conditions. A consistent multi-objective optimization procedure is also applied. The function introduced here consists of minimizing the weight and cost, as well as of maximizing the load. A genetic algorithm is used for the optimization process.     

Efficient methods using high accuracy approximate inertial manifolds

Summary. We extend the idea of the post-processing Galerkin method, in the context of dissipative evolution equations, to the nonlinear Galerkin, the filtered Galerkin, and the filtered nonlinear Galerkin methods. In general, the post-processing algorithm takes advantage of the fact that the error committed in the lower modes of the nonlinear Galerkin method (and Galerkin method), for approximating smooth, bounded solutions, is much smaller than the total error of the method. In each case, an improvement in accuracy is obtained by post-processing these more accurate lower modes with an appropriately chosen, highly accurate, approximate inertial manifold (AIM). We present numerical experiments that support the theoretical improvements in accuracy. Both the theory and computations are presented in the framework of a two dimensional reaction-diffusion system with polynomial nonlinearity. However, the algorithm is very general and can be implemented for other dissipative evolution systems. The computations clearly show the post-processed filtered Galerkin method to be the most efficient method. Received September 10, 1998 / Revised version received April 26, 1999 / Published online July 12, 2000     

Application of spectral filtering to discontinuous Galerkin methods on triangulations

We adapt the spectral viscosity (SV) formulation implemented as a modal filter to a discontinuous Galerkin (DG) method solving hyperbolic conservation laws on triangular grids. The connection between SV and spectral filtering, which is undertaken for the first time in the context of DG methods on unstructured grids, allows to specify conditions on the filter strength regarding time step choice and mesh refinement. A crucial advantage of this novel damping strategy is its low computational cost. We furthermore obtain new error bounds for filtered Dubiner expansions of smooth functions. While high order accuracy with respect to the polynomial degree N is proven for the filtering procedure in this case, an adaptive application is proposed to retain the high spatial approximation order. Although spectral filtering stabilizes the scheme, it leaves weaker oscillations. Therefore, as a postprocessing step, we apply the image processing technique of digital total variation (DTV) filtering in the new context of DG solutions and prove conservativity in the limit for this filtering procedure. Numerical experiments for scalar conservation laws confirm the designed order of accuracy of the DG scheme with adaptive modal filtering for polynomial degrees up to 8 and the viability of spectral and DTV filtering in case of shocks. © 2011 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2011