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     

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     

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     

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     

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     

New a priori analysis of first‐order system least‐squares finite element methods for parabolic problems

We provide new insights into the a priori theory for a time‐stepping scheme based on least‐squares finite element methods for parabolic first‐order systems. The elliptic part of the problem is of general reaction‐convection‐diffusion type. The new ingredient in the analysis is an elliptic projection operator defined via a nonsymmetric bilinear form, although the main bilinear form corresponding to the least‐squares functional is symmetric. This new operator allows to prove optimal error estimates in the natural norm associated to the problem and, under additional regularity assumptions, in the L² norm. Numerical experiments are presented which confirm our theoretical findings.     

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)     

Error analysis of the L2 least‐squares finite element method for incompressible inviscid rotational flows

In this article we analyze the L² least‐squares finite element approximations to the incompressible inviscid rotational flow problem, which is recast into the velocity‐vorticity‐pressure formulation. The least‐squares functional is defined in terms of the sum of the squared L² norms of the residual equations over a suitable product function space. We first derive a coercivity type a priori estimate for the first‐order system problem that will play the crucial role in the error analysis. We then show that the method exhibits an optimal rate of convergence in the H¹ norm for velocity and pressure and a suboptimal rate of convergence in the L² norm for vorticity. A numerical example in two dimensions is presented, which confirms the theoretical error estimates. © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2004     

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.     

Split least‐squares finite element methods for non‐Fickian flow in porous media

In this article, we introduce two least‐squares finite element procedures for parabolic integro‐differential equations arising in the modeling of non‐Fickian flow in porous media. By selecting the least‐squares functional properly the presented procedure can be split into two independent subprocedures, one subprocedure is for the primitive unknown and the other is for the flux. The optimal order convergence analysis is established. Numerical examples are given to show the efficiency of the introduced schemes. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013     

On a mixed finite element formulation of a second‐order quasilinear problem in the plane

We analyze a mixed finite element discretization of a second‐order quasilinear problem based on the Raviart‐Thomas space. We prove that the discrete problem is solvable and provide a local uniqueness result for the solution. We also obtain optimal order L²‐error estimates for both the scalar variable and the associated flux. The main feature of our method is that it is free from the boundness conditions required in previous works on the coefficients of the quasilinear operator. © 2003 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 20: 90–103, 2004.     

Singular solutions of the Poisson equation at edges of three‐dimensional domains and their treatment with a predictor–corrector finite element method

Solutions of boundary value problems in three‐dimensional domains with edges may exhibit singularities which are known to influence both the accuracy of the finite element solutions and the rate of convergence in the error estimates. This paper considers boundary value problems for the Poisson equation on typical domains Ω ? ?³ with edge singularities and presents, on the one hand, explicit computational formulas for the flux intensity functions. On the other hand, it proposes and analyzes a nonconforming finite element method on regular meshes for the efficient treatment of the singularities. The novelty of the present method is the use of the explicit formulas for the flux intensity functions in defining a postprocessing procedure in the finite element approximation of the solution. A priori error estimates in H¹(Ω) show that the present algorithm exhibits the same rate of convergence as it is known for problems with regular solutions.     

Analysis of a two‐stage least‐squares finite element method for the planar elasticity problem

A new first‐order formulation for the two‐dimensional elasticity equations is proposed by introducing additional variables which, called stresses here, are the derivatives of displacements. The resulted stress–displacement system can be further decomposed into two dependent subsystems, the stress system and the displacement system recovered from the stresses. For constructing finite element approximations to these subsystems with appropriate boundary conditions, a two‐stage least‐squares procedure is introduced. The analysis shows that, under suitable regularity assumptions, the rates of convergence of the least‐squares approximations for all the unknowns are optimal both in the H¹‐norm and in L²‐norm. Also, numerical experiments with various Poisson's ratios are examined to demonstrate the theoretical estimates. Copyright © 1999 John Wiley & Sons, Ltd.     

Least‐squares method for the Oseen equation

This article studies the least‐squares finite element method for the linearized, stationary Navier–Stokes equation based on the stress‐velocity‐pressure formulation in d dimensions (d = 2 or 3). The least‐squares functional is simply defined as the sum of the squares of the L² norm of the residuals. It is shown that the homogeneous least‐squares functional is elliptic and continuous in the norm. This immediately implies that the a priori error estimate of the conforming least‐squares finite element approximation is optimal in the energy norm. The L² norm error estimate for the velocity is also established through a refined duality argument. Moreover, when the right‐hand side f belongs only to , we derive an a priori error bound in a weaker norm, that is, the norm. © 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 1289–1303, 2016     

Nonconforming finite element methods with subgrid viscosity applied to advection‐diffusion‐reaction equations

A nonconforming (Crouzeix–Raviart) finite element method with subgrid viscosity is analyzed to approximate advection‐diffusion‐reaction equations. The error estimates are quasi‐optimal in the sense that keeping the Péclet number fixed, the estimates are suboptimal of order in the mesh size for the L²‐norm and optimal for the advective derivative on quasi‐uniform meshes. The method is also reformulated as a finite volume box scheme providing a reconstruction formula for the diffusive flux with local conservation properties. Numerical results are presented to illustrate the error analysis. © 2006 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2006     

A remark on split least-squares mixed element procedures for pseudo-parabolic equations

In this paper, we introduce two novel split least-squares mixed element procedures for pseudo-parabolic equations. By selecting the least-squares functional properly, each procedure can be split into two independent symmetric positive definite sub-procedures. One of sub-procedures is for the primitive unknown variable u, which is the same as the standard Galerkin finite element procedure and the other is for the introduced flux variable σ. Optimal order error estimates are developed. A numerical example is given to show the efficiency of the introduced schemes.     

Superconvergent error estimates for a class of discretization methods for a coupled first‐order system with discontinuous coefficients

Lithological discontinuities in a reservoir generate discontinuous coefficients for the first‐order system of equations used in the simulation of fluid flow in porous media. Systems of conservation laws with discontinuous coefficients also arise in many other physical applications. In this article, we present a class of discretization schemes that include variants of mixed finite element methods, finite volume element methods, and cell‐centered finite difference equations as special cases. Error estimates of the order O(h²) in certain discrete L²‐norms are established for both the primary independent variable and its flux, even in the presence of discontinuous coefficients in the flux term. © 1999 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 15: 267–283, 1999     

Methods for the numerical solution of the Benjamin‐Bona‐Mahony‐Burgers equation

L^∞‐error estimates for finite element for Galerkin solutions for the Benjamin‐Bona‐Mahony‐Burgers (BBMB) equation are considered. A priori bound and the semidiscrete Galerkin scheme are studied using appropriate projections. For fully discrete Galerkin schemes, we consider the backward Euler method and analyze the corresponding error estimates. For a second order accuracy in time, we propose a three‐level backward method. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2008     

Normal and tangential continuous elements for least‐squares mixed finite element methods

We consider a finite element discretization of the primal first‐order least‐squares mixed formulation of the second‐order elliptic problem. The unknown variables are displacement and flux, which are approximated by equal‐order elements of the usual continuous element and the normal continuous element, respectively. We show that the error bounds for all variables are optimal. In addition, a field‐based least‐squares finite element method is proposed for the 3D‐magnetostatic problem, where both magnetic field and magnetic flux are taken as two independent variables which are approximated by the tangential continuous and the normal continuous elements, respectively. Coerciveness and optimal error bounds are obtained. © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2004.     

Robust residual‐ and recovery‐based a posteriori error estimators for a multipoint flux mixed finite element discretization of interface problems

We consider a posteriori error estimation for a multipoint flux mixed finite element method for two‐dimensional elliptic interface problems. Within the class of modified quasi‐monotonically distributed coefficients, we derive a residual‐type a posteriori error estimator of the weighted sum of the scalar and flux errors which is robust with respect to the jumps of the coefficients. Moreover, we develop robust implicit and explicit recovery‐type estimators through gradient recovery in an H(curl)‐conforming finite element space. In particular, we apply a modified L² projection in the implicit recovery procedure so as to reduce the computational cost of the recovered gradient. Numerical experiments confirm the theoretical results.