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     

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     

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     

Reconstructed Discontinuous Approximation to Stokes Equation in a Sequential Least Squares Formulation

We propose a new least squares finite element method to solve the Stokes problem with two sequential steps. The approximation spaces are constructed by the patch reconstruction with one unknown per element. For the first step, we reconstruct an approximation space consisting of piecewise curl-free polynomials with zero trace. By this space, we minimize a least squares functional to obtain the numerical approximations to the gradient of the velocity and the pressure. In the second step, we minimize another least squares functional to give the solution to the velocity in the reconstructed piecewise divergence-free space. We derive error estimates for all unknowns under both $L^2$ norms and energy norms. Numerical results in two dimensions and three dimensions verify the convergence rates and demonstrate the great flexibility of our method.     

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     

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.     

Remarks on a Least Squares Mixed Finite Element for Plane Elasticity

The objective of this work is to discuss a least squares finite element method within plane elasticity problems. The L ₂-norm minimization of the residuals of the given first order system of differential equations leads to a functional, which is a two field formulation in the displacements and the stresses. The governing equations for the considered least squares mixed finite element are derived. (© 2006 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     

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.     

Error estimates of expanded mixed methods for optimal control problems governed by hyperbolic integro‐differential equations

In this article, we investigate the L^∞(L²) ‐error estimates of the semidiscrete expanded mixed finite element methods for quadratic optimal control problems governed by hyperbolic integrodifferential equations. The state and the costate are discretized by the order k Raviart‐Thomas mixed finite element spaces, and the control is approximated by piecewise polynomials of order k(k ≥ 0). We derive error estimates for both the state and the control approximation. Numerical experiments are presented to test the theoretical results. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013     

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.     

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     

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     

Finite element differential forms

A differential form is a field which assigns to each point of a domain an alternating multilinear form on its tangent space. The exterior derivative operation, which maps differential forms to differential forms of the next higher order, unifies the basic first order differential operators of calculus, and is a building block for a great variety of differential equations. When discretizing such differential equations by finite element methods, stable discretization depends on the development of spaces of finite element differential forms. As revealed recently through the finite element exterior calculus, for each order of differential form, there are two natural families of finite element subspaces associated to a simplicial triangulation. In the case of forms of order zero, which are simply functions, these two families reduce to one, which is simply the well-known family of Lagrange finite element subspaces of the first order Sobolev space. For forms of degree 1 and of degree n − 1 (where n is the space dimension), we obtain two natural families of finite element subspaces, unifying many of the known mixed finite element spaces developed over the last decades. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Semidiscrete least squares methods for linear hyperbolic systems

Some approximate methods for solving linear hyperbolic systems are presented and analyzed. The methods consist of discretizing with respect to time and solving the resulting hyperbolic system for fixed time by least squares finite element methods. An analysis of least squares approximations is given, including optimal order estimates for piecewise polynomial approximation spaces. Numerical results for the inviscid Burgers' equation are also presented. © 1992 John Wiley & Sons, Inc.     

The structured total least‐squares approach for non‐linearly structured matrices

In this paper, an extension of the structured total least‐squares (STLS) approach for non‐linearly structured matrices is presented in the so‐called ‘Riemannian singular value decomposition’ (RiSVD) framework. It is shown that this type of STLS problem can be solved by solving a set of Riemannian SVD equations. For small perturbations the problem can be reformulated into finding the smallest singular value and the corresponding right singular vector of this Riemannian SVD. A heuristic algorithm is proposed. Some examples of Vandermonde‐type matrices are used to demonstrate the improved accuracy of the obtained parameter estimator when compared to other methods such as least squares (LS) or total least squares (TLS). Copyright © 2002 John Wiley & Sons, Ltd.     

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     

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     

Error estimates for mixed finite element approximations of the viscoelasticity wave equation

This paper studies mixed finite element approximations to the solution of the viscoelasticity wave equation. Two new transformations are introduced and a corresponding system of first‐order differential‐integral equations is derived. The semi‐discrete and full‐discrete mixed finite element methods are then proposed for the problem based on the Raviart–Thomas–Nedelec spaces. The optimal error estimates in L²‐norm are obtained for the semi‐discrete and full‐discrete mixed approximations of the general viscoelasticity wave equation. Copyright © 2004 John Wiley & Sons, Ltd.