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)     

Least‐squares mixed finite element methods for the incompressible Navier‐Stokes equations

Least‐squares mixed finite element schemes are formulated to solve the evolutionary Navier‐Stokes equations and the convergence is analyzed. We recast the Navier‐Stokes equations as a first‐order system by introducing a vorticity flux variable, and show that a least‐squares principle based on L² norms applied to this system yields optimal discretization error estimates. © 2002 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 18: 441–453, 2002; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/num.10015     

A least‐squares finite element approximation for the compressible Stokes equations

This article studies a least‐squares finite element method for the numerical approximation of compressible Stokes equations. Optimal order error estimates for the velocity and pressure in the H¹ are established. The choice of finite element spaces for the velocity and pressure is not subject to the inf‐sup condition. © 2000 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 16: 62–70, 2000     

An analysis for the compressible Stokes equations by first‐order system of least‐squares finite element method

This article applies the first‐order system least‐squares (fosls) finite element method developed by Cai, Manteuffel and McCormick to the compressible Stokes equations. By introducing a new dependent velocity flux variable, we recast the compressible Stokes equations as a first‐order system. Then it is shown that the ellipticity and continuity hold for the least‐squares functionals employing the mixture of H^?1 and L², so that the fosls finite element methods yield best approximations for the velocity flux and velocity. © 2001 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 17:689–699, 2001     

Superconvergence of the split least‐squares method for second‐order hyperbolic equations

This article studies superconvergence phenomena of the split least‐squares mixed finite element method for second‐order hyperbolic equations. By selecting the least‐squares functional properly, the procedure can be split into two independent symmetric positive definite subprocedures, one of which is for the primitive unknown and the other is for the flux. Based on interpolation operators and an auxiliary projection, superconvergent H¹ error estimates for the primary variable u and L² error estimates for the introduced flux variable σ are obtained under the standard quasiuniform assumptions on finite element partition. A numerical example is given to show the performance of the introduced scheme. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 222‐238, 2014     

On an implicit particle method for simulation of forming processes

Task is the simulation of forming processes using particle methods. We implemented some mesh-free methods (the element free Galerkin method [1] and others) and the finite element method in one programme system which permits a direct comparison. For the mesh-free methods a moving least squares approximation is applied. The shape functions are not zero or one at the nodes, thus essential boundary conditions cannot be imposed directly [2]. We use a penalty method to enforce essential boundary conditions and contact conditions. The contact algorithm (normal contact of nodes to C¹-continuous surfaces) is checked by means of the element free Galerkin method and the FEM on the basis of numerical examples which deal with forming processes. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Optimal Error Estimate of the Penalty Finite Element Method for the Micropolar Fluid Equations

We present an optimal error estimate of the numerical velocity, pressure, and angular velocity for the fully discrete penalty finite element method of the micropolar equations when the parameters ?, Δ t, and h are sufficiently small. In order to obtain this estimate, we present the time discretization of the penalty micropolar equation that is based on the backward Euler scheme; the spatial discretization of the time discretized penalty micropolar equation is based on a finite elements space pair (X _h , M _h ) that satisfies some approximations properties.     

Mortar element methods for parabolic problems

In this article a standard mortar finite element method and a mortar element method with Lagrange multiplier are used for spatial discretization of a class of parabolic initial‐boundary value problems. Optimal error estimates in L^∞(L²) and L^∞(H¹)‐norms for semidiscrete methods for both the cases are established. The key feature that we have adopted here is to introduce a modified elliptic projection. In the standard mortar element method, a completely discrete scheme using backward Euler scheme is discussed and optimal error estimates are derived. The results of numerical experiments support the theoretical results obtained in this article. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2008     

Two grid finite element discretization method for semi‐linear hyperbolic integro‐differential equations

In this paper, we will investigate a two grid finite element discretization method for the semi‐linear hyperbolic integro‐differential equations by piecewise continuous finite element method. In order to deal with the semi‐linearity of the model, we use the two grid technique and derive that once the coarse and fine mesh sizes H, h satisfy the relation h = H² for the two‐step two grid discretization method, the two grid method achieves the same convergence accuracy as the ordinary finite element method. Both theoretical analysis and numerical experiments are given to verify the results.     

An Adaptive Local (AL) Basis for Elliptic Problems with Complicated Discontinuous Coefficients

We develop a generalized finite element method for the discretization of elliptic partial differential equations in heterogeneous media. In [5] a semidiscrete method has been introduced to set up an adaptive local finite element basis (AL basis) on a coarse mesh with mesh size H which, typically, does not resolve the matrix of the media while the textbook finite element convergence rates are preserved. This method requires O(log(1/H)^d+1) basis functions per mesh point where d denotes the spatial dimension of the computational domain. We present a fully discrete version of this method, where the AL basis is constructed by solving finite-dimensional localized problems, and which preserves the optimal convergence rates. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the finite volume element method

Summary The finite volume element method (FVE) is a discretization technique for partial differential equations. It uses a volume integral formulation of the problem with a finite partitioning set of volumes to discretize the equations, then restricts the admissible functions to a finite element space to discretize the solution. this paper develops discretization error estimates for general selfadjoint elliptic boundary value problems with FVE based on triangulations with linear finite element spaces and a general type of control volume. We establishO(h) estimates of the error in a discreteH ¹ semi-norm. Under an additional assumption of local uniformity of the triangulation the estimate is improved toO(h ²). Results on the effects of numerical integration are also included.This research was sponsored in part by the Air Force Office of Scientific Research under grant number AFOSR-86-0126 and the National Science Foundation under grant number DMS-8704169. This work was performed while the author was at the University of Colorado at Denver     

Convergence of finite element method for linear second‐order wave equations with discontinuous coefficients

A finite element method is proposed and analyzed for hyperbolic problems with discontinuous coefficients. The main emphasize is given on the convergence of such method. Due to low global regularity of the solutions, the error analysis of the standard finite element method is difficult to adopt for such problems. For a practical finite element discretization, optimal error estimates in L^∞(L²) and L^∞(H¹) norms are established for continuous time discretization. Further, a fully discrete scheme based on a symmetric difference approximation is considered, and optimal order convergence in L^∞(H¹) norm is established. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013     

Nonlinear Galerkin method and two-step method for the Navier-Stokes equations

This article represents a new nonlinear Galerkin scheme for the Navier-Stokes equations. This scheme consists of a nonlinear Galerkin finite element method and a two-step difference method. Moreover, we also provide a Galerkin scheme. By convergence analysis, two numerical schemes have the same second-order convergence accuracy for the spatial discretization and time discretization if H is chosen such that H = O(h^2/3). However, the nonlinear Galerkin scheme is simpler than the Galerkin scheme, namely, this scheme can save a large amount of computational time. © 1996 John Wiley & Sons, Inc.     

On the subdomain‐Galerkin/least squares method for 2‐ and 3‐D mixed elliptic problems with reaction terms

In this article we apply the subdomain‐Galerkin/least squares method, which is first proposed by Chang and Gunzburger for first‐order elliptic systems without reaction terms in the plane, to solve second‐order non‐selfadjoint elliptic problems in two‐ and three‐dimensional bounded domains with triangular or tetrahedral regular triangulations. This method can be viewed as a combination of a direct cell vertex finite volume discretization step and an algebraic least‐squares minimization step in which the pressure is approximated by piecewise linear elements and the flux by the lowest order Raviart‐Thomas space. This combined approach has the advantages of both finite volume and least‐squares methods. Among other things, the combined method is not subject to the Ladyzhenskaya‐Babus?ka‐Brezzi condition, and the resulting linear system is symmetric and positive definite. An optimal error estimate in the H¹(Ω) × H(div; Ω) norm is derived. An equivalent residual‐type a posteriori error estimator is also given. © 2002 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 18: 738–751, 2002; Published online in Wiley InterScience (www.interscience.wiley.com); DOI 10.1002/num.10030.     

Convergence of the interpolated coefficient finite element method for the two‐dimensional elliptic sine‐Gordon equations

An interpolated coefficient finite element method is presented and analyzed for the two‐dimensional elliptic sine‐Gordon equations with Dirichlet boundary conditions. It is proved that the discretization scheme admits at least one solution, and that a subsequence of the approximation solutions converges to an exact solution in L²‐norm as the mesh size tends to zero. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2011     

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     

Least‐squares mixed finite element methods for the RLW equations

A least‐squares mixed finite element (LSMFE) schemes are formulated to solve the 1D regularized long wave (RLW) equations and the convergence is discussed. The L² error estimates of LSMFE methods for RLW equations under the standard regularity assumption on the finite element partition are given.© 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2008     

Negative norm least‐squares methods for the velocity‐vorticity‐pressure Navier–Stokes equations

We develop and analyze a least‐squares finite element method for the steady state, incompressible Navier–Stokes equations, written as a first‐order system involving vorticity as new dependent variable. In contrast to standard L² least‐squares methods for this system, our approach utilizes discrete negative norms in the least‐squares functional. This allows us to devise efficient preconditioners for the discrete equations, and to establish optimal error estimates under relaxed regularity assumptions. © 1999 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 15: 237–256, 1999     

Incomplete block-matrix factorization iterative methods for convection-diffusion problems

Standard Galerkin finite element methods or finite difference methods for singular perturbation problems lead to strongly unsymmetric matrices, which furthermore are in general notM-matrices. Accordingly, preconditioned iterative methods such as preconditioned (generalized) conjugate gradient methods, which have turned out to be very successful for symmetric and positive definite problems, can fail to converge or require an excessive number of iterations for singular perturbation problems.This is not so much due to the asymmetry, as it is to the fact that the spectrum can have both eigenvalues with positive and negative real parts, or eigenvalues with arbitrary small positive real parts and nonnegligible imaginary parts. This will be the case for a standard Galerkin method, unless the meshparameterh is chosen excessively small. There exist other discretization methods, however, for which the corresponding bilinear form is coercive, whence its finite element matrix has only eigenvalues with positive real parts; in fact, the real parts are positive uniformly in the singular perturbation parameter.In the present paper we examine the streamline diffusion finite element method in this respect. It is found that incomplete block-matrix factorization methods, both on classical form and on an inverse-free (vectorizable) form, coupled with a general least squares conjugate gradient method, can work exceptionally well on this type of problem. The number of iterations is sometimes significantly smaller than for the corresponding almost symmetric problem where the velocity field is close to zero or the singular perturbation parameter =1.The 2 ^nd author's research was sponsored by Control Data Corporation through its PACER fellowship program.The 3 ^rd author's research was supported by the Netherlands organization for scientific research (NWO).On leave from the Institute of Mathematics, Academy of Science, 1090 Sofia, P.O. Box 373, Bulgaria.     

Error estimate of the P 1 nonconforming finite element method for the penalized unsteady Navier-Stokes equations

We consider a finite element method for the penalty formulation of the time dependent Navier-Stokes equations. Usually the improper choice of the finite element space will lead that the error estimate (inversely) depends on the penalty parameter e{\epsilon}. We use the classical P ₁ nonconforming finite element space for the spatial discretization. Optimal e{\epsilon}-uniform error estimations for both velocity and pressure are obtained.