An adaptive least squares mixed finite element method for the stress‐displacement formulation of linear elasticity

A least‐squares mixed finite element method for linear elasticity, based on a stress‐displacement formulation, is investigated in terms of computational efficiency. For the stress approximation quadratic Raviart‐Thomas elements are used and these are coupled with the quadratic nonconforming finite element spaces of Fortin and Soulie for approximating the displacement. The local evaluation of the least‐squares functional serves as an a posteriori error estimator to be used in an adaptive refinement algorithm. We present computational results for a benchmark test problem of planar elasticity including nearly incompressible material parameters in order to verify the effectiveness of our adaptive strategy. For comparison, conforming quadratic finite elements are also used for the displacement approximation showing convergence orders similar to the nonconforming case, which are, however, not independent of the Lamé parameters. © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2005     

Option pricing with a direct adaptive sparse grid approach

We present an adaptive sparse grid algorithm for the solution of the Black–Scholes equation for option pricing, using the finite element method. Sparse grids enable us to deal with higher-dimensional problems better than full grids. In contrast to common approaches that are based on the combination technique, which combines different solutions on anisotropic coarse full grids, the direct sparse grid approach allows for local adaptive refinement. When dealing with non-smooth payoff functions, this reduces the computational effort significantly. In this paper, we introduce the spatially adaptive discretization of the Black–Scholes equation with sparse grids and describe the algorithmic structure of the numerical solver. We present several strategies for adaptive refinement, evaluate them for different dimensionalities, and demonstrate their performance showing numerical results.     

Least squares moving finite elements

The moving finite element (MFE) method, when applied to purelyhyperbolic partial differential equation, moves nodes with approximatelycharacteristic speeds, which makes the method useless for steady-stateproblems. We introduce the least squares MFE method (LSMFE)for steady-state pure convection problems which corrects thisdefect. We show results for a steady-state pure convection problemin one dimension in which the nodes are no longer swept downstreamas in MFE. The method is then extended to two dimensions andthe grid aligns automatically with the flow, thereby yieldingfar greater accuracy than the corresponding fixed node leastsquares results, as is shown in two-dimensional numerical trials.     

On the convergence of nonnested multigrid methods with nested spaces on coarse grids

Multigrid methods for discretized partial differential problems using nonnested conforming and nonconforming finite elements are here defined in the general setting. The coarse‐grid corrections of these multigrid methods make use of different finite element spaces from those on the finest grid. In general, the finite element spaces on the finest grid are nonnested, while the spaces are nested on the coarse grids. An abstract convergence theory is developed for these multigrid methods for differential problems without full elliptic regularity. This theory applies to multigrid methods of nonnested conforming and nonconforming finite elements with the coarse‐grid corrections established on nested conforming finite element spaces. Uniform convergence rates (independent of the number of grid levels) are obtained for both the V and W‐cycle methods with one smoothing on all coarse grids and with a sufficiently large number of smoothings solely on the finest grid. In some cases, these uniform rates are attained even with one smoothing on all grids. The present theory also applies to multigrid methods for discretized partial differential problems using mixed finite element methods. © 2000 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 16: 265–284, 2000     

An adaptive wavelet viscosity method for hyperbolic conservation laws

We extend the multiscale finite element viscosity method for hyperbolic conservation laws developed in terms of hierarchical finite element bases to a (pre‐orthogonal spline‐)wavelet basis. Depending on an appropriate error criterion, the multiscale framework allows for a controlled adaptive resolution of discontinuities of the solution. The nonlinearity in the weak form is treated by solving a least‐squares data fitting problem. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2008     

Front tracking for two-phase flow in reservoir simulation by adaptive mesh

In this article, an algorithm for the numerical approximation of two-phase flow in porous media by adaptive mesh is presented. A convergent and conservative finite volume scheme for an elliptic equation is proposed, together with the finite difference schemes, upwind and MUSCL, for a hyperbolic equation on grids with local refinement. Hence, an IMPES method is applied in an adaptive composite grid to track the front of a moving solution. An object-oriented programmation technique is used. The computational results for different examples illustrate the efficiency of the proposed algorithm. © 1997 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 13: 673–697, 1997     

Two‐grid mixed finite‐element methods for nonlinear Schrödinger equations

Two‐grid mixed finite element schemes are developed for solving both steady state and unsteady state nonlinear Schrödinger equations. The schemes use discretizations based on a mixed finite‐element method. The two‐grid approach yields iterative procedures for solving the nonlinear discrete equations. The idea is to relegate all of the Newton‐like iterations to grids much coarser than the final one, with no loss in order of accuracy. Numerical tests are performed. © 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 28: 63‐73, 2012     

The Finite Difference Based Fast Adaptive Composite Grid Method

The fast adaptive composite grid (FAC) method is an iterative method for solving discrete boundary value problems on composite grids. McCormick introduced the method in [8] and considered the convergence behaviour for discrete problems resulting from finite volume element discretization on composite grids. In this paper we consider discrete problems resulting from finite difference discretization on composite grids. We distinguish between two obvious discretization approaches at the grid points on the interfaces between fine and coarse subgrids. The FAC method for solving such discrete problems is described. In the FAC method several intergrid transfer operators appear. We study how the convergence behaviour depends on these intergrid transfer operators. Based on theoretical insights, (quasi-)optimal intergrid transfer operators are derived. Numerical results illustrate the fast convergence of the FAC method using these intergrid transfer operators.     

An adaptive algorithm in time domain for dynamic analysis of a simply supported beam subjected to a moving vehicle

This paper presents an adaptive algorithm in the time domain for the dynamic analysis of a simply supported beam subjected to the moving load and moving vehicle with/without varying surface roughness. By expanding variables at a discretized time interval, a coupled spatial‐temporal problem can be converted into a series of recursive space problems that are solved by finite element method (FEM), and a piecewised adaptive computing procedure can be carried out for different sizes of time steps. The proposed approach is numerically verified via the comparison with analytical and the Runge–Kutta method‐based solutions, and satisfactory results have been achieved. Copyright © 2011 John Wiley & Sons, Ltd.     

Adaptive finite element approximations on nonmatching grids for second‐order elliptic problems

In this article we consider a finite element approximation for a model elliptic problem of second order on non‐matching grids. This method combines the continuous finite element method with interior penalty discontinuous Galerkin method. As a special case, we develop a finite element method that is continuous on the matching part of the grid and is discontinuous on the nonmatching part. A residual type a posteriori error estimate is derived. Results of numerical experiments are presented. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2010     

抛物型界面问题的变网格有限元方法

本文针对抛物型界面问题,提出了一种线性三角形变网格有限元方法.其主要思路是针对空间变量采用有限元离散,对时间变量采用差分离散,但是不同时刻的有限元剖分网格可以不同.在不引入Ritz投影这一传统分析工具的情况下,得到了最优误差估计结果,使得证明过程更加简洁.给出的数值算例验证了理论分析的正确性.     

非定常的热传导-对流问题的非线性Galerkin混合元法(Ⅱ):向后一步的Euler全离散化格式

In this paper, a fully discrete format of nonlinear Galerkin mixed element method with backward one-step Euler discretization of time for the non stationary conduction-convection problems is presented. The scheme is based on two finite element spaces XH and Xh for the approximation of the velocity, defined respectively on a coarse grid with grids size H and another fine grid with grid size h<< H, a finite element space Mh for the approximation of the pressure and two finite element spaces AH and Wh, for the approximation of the temperature,also defined respectivply on the coarse grid with grid size H and another fine grid with grid size h. The existence and the convergence of the fully discrete mixed element solution are shown. The scheme consists in using standard backward one step Euler-Galerkin fully discrete format at first L0 steps (L0 2) on fine grid with grid size h, but using nonlinear Galerkin mixed element method of backward one step Euler-Galerkin fully discrete format through L0 + 1 step to end step. We have proved that the fully discrete nonlinear Galerkin mixed element procedure with respect to the coarse grid spaces with grid size H holds superconvergence.     

A Least‐Squares Formulation for the Shallow Water Equations with Small Viscosity

A least‐squares finite element method for the shallow water equations with viscosity parameter μ > 0 is proposed and studied. The shallow water equations are reformulated as a first order system by adding a new variable for the velocity flux. The reformulated first order system is combined with a characteristic‐based time discretization and a least squares approach. For the correct boundary treatment in the limit case μ → 0, a trace theorem is presented. For the numerical computation of the velocity, the finite element spaces introduced recently by Mardal, Tai and Winther (SIAM Journal on Numerical Analysis 40, pp. 1605–1631) are used. The degrees of freedom in these spaces can be identified with the normal and tangential components, respectively. Numerical results for some test examples are shown. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

The analytical solution of two interesting hyperbolic problems as a test case for a finite volume method with a new grid refinement technique

A finite volume method with grid adaption is applied to two hyperbolic problems: the ultra-relativistic Euler equations, and a scalar conservation law. Both problems are considered in two space dimensions and share the common feature of moving shock waves. In contrast to the classical Euler equations, the derivation of appropriate initial conditions for the ultra-relativistic Euler equations is a non-trivial problem that is solved using one-dimensional shock conditions and the Lorentz invariance of the system. The discretization of both problems is based on a finite volume method of second order in both space and time on a triangular grid. We introduce a variant of the min-mod limiter that avoids unphysical states for the Euler system. The grid is adapted during the integration process. The frequency of grid adaption is controlled automatically in order to guarantee a fine resolution of the moving shock fronts. We introduce the concept of “width refinement” which enlarges the width of strongly refined regions around the shock fronts; the optimal width is found by a numerical study. As a result we are able to improve efficiency by decreasing the number of adaption steps. The performance of the finite volume scheme is compared with several lower order methods.     

A shock indicator for adaptive transonic flow computations

Summary An adaptive finite element method for the calculation of transonic potential flows was developed. A residual based error indicator is complemented by a shock indicator. For a good shock resolution mesh refinement as well as moving nodes were needed. An analysis of the method and computational results are given.The research reported in this article was supported by the Deutsche Forschungsgemeinschaft and the Volkswagen-Stiftung     

An optimal‐order estimate for MMOC‐MFEM approximations to porous medium flow

Mathematical models used to describe porous medium flow lead to coupled systems of time‐dependent partial differential equations. Standard methods tend to generate numerical solutions with nonphysical oscillations or numerical dispersion along with spurious grid‐orientation effect. The MMOC‐MFEM time‐stepping procedure, in which the modified method of characteristics (MMOC) is used to solve the transport equation and a mixed finite element method (MFEM) is used for the pressure equation, simulates porous medium flow accurately even if large spatial grids and time steps are used. In this article we prove an optimal‐order error estimate for a family of MMOC‐MFEM approximations. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009     

A comparative analysis of adaptive algorithms in the finite element method for solving the boundary value problem for a stationary reaction-diffusion equation

A new adaptive algorithm is proposed for constructing grids in the hp-version of the finite element method with piecewise polynomial basis functions. This algorithm allows us to find a solution (with local singularities) to the boundary value problem for a one-dimensional reaction-diffusion equation and smooth the grid solution via the adaptive elimination and addition of grid nodes. This algorithm is compared to one proposed earlier that adaptively refines the grid and deletes nodes with the help of an estimate for the local effect of trial addition of new basis functions and the removal of old ones. Results are presented from numerical experiments aimed at assessing the performance of the proposed algorithm on a singularly perturbed model problem with a smooth solution.     

Grid construction for discretely defined configurations

An important element of global software codes for computing real-life three-dimensional problems with singularities (such as boundary and internal layers, shocks, detonation waves, combustion fronts, high-speed jets, and phase transition zones) is automatic adaptive grid generation, which can considerably enhance the efficiency of computer resource management. In three-dimensional domains with boundaries of complex geometry, in particular, with discretely defined boundaries, adaptive grids are generated by applying inverted Beltrami and diffusion equations for a spherical monitor tensor.     

An a posteriori error estimator for anisotropic refinement

Summary. Besides an algorithm for local refinement, an a posteriori error estimator is the basic tool of every adaptive finite element method. Using information generated by such an error estimator the refinement of the grid is controlled. For 2nd order elliptic problems we present an error estimator for anisotropically refined grids, like -d cuboidal and 3-d prismatic grids, that gives correct information about the size of the error; additionally it generates information about the direction into which some element has to be refined to reduce the error in a proper way. Numerical examples are presented for 2-d rectangular and 3-d prismatic grids. Received March 15, 1994 / Revised version received June 3, 1994     

A note on the least squares solution of the groundwater flow equation

The least squares finite element method is a member of the weighted residuals class of numerical methods for solving partial differential equations. The least squares finite element method is applied to the groundwater flow equation. Space is discretized with a C¹ continuous trial function and parameters are approximated with a C⁰ bilinear basis. Solutions for problems containing parameters with large localized spatial gradients are characterized by errors that are propagated throughout the entire domain. Second-order spatial convergence is observed, and extreme mesh refinement is required to match Galerkin and mixed least squares finite element results. Temporal discretization should be kept separate from the least squares spatial discretization. © 1994 John Wiley & Sons, Inc.