New formulation of iterative substructuring methods without Lagrange multipliers: Neumann–Neumann and FETI

This article is devoted to introduce a new approach to iterative substructuring methods that, without recourse to Lagrange multipliers, yields positive definite preconditioned formulations of the Neumann–Neumann and FETI types. To my knowledge, this is the first time that such formulations have been made without resource to Lagrange multipliers. A numerical advantage that is concomitant to such multipliers‐free formulations is the reduction of the degrees of freedom associated with the Lagrange multipliers. Other attractive features are their generality, directness, and simplicity. The general framework of the new approach is rather simple and stems directly from the discretization procedures that are applied; in it, the differential operators act on discontinuous piecewise‐defined functions. Then, the Lagrange multipliers are not required because in such an environment the functions‐discontinuities are not an anomaly that need to be corrected. The resulting algorithms and equations‐systems are also derived with considerable detail. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2008     

Multipliers‐free dual‐primal domain decomposition methods for nonsymmetric matrices and their numerical testing

The most commonly used nonoverlapping domain decomposition algorithms, such as the FETI‐DP and BDDC, require the introduction of discontinuous vector spaces. Most of the works on such methods are based on approaches that originated in Lagrange multipliers formulations. Using a theory of partial differential equations formulated in discontinuous piecewise‐defined functions, introduced and developed by Herrera and his collaborators through a long time span, recently the authors have developed an approach to domain decomposition methods in which general problems with prescribed jumps are treated at the discrete level. This yields an elegant and general direct framework that permits analyzing the problems in greater detail. The algorithms derived using it have properties similar to those of well‐established methods such as FETI‐DP, but, in our experience, they are easier to implement. Also, they yield explicit matrix formulas that unify the different methods. Furthermore, this multipliers‐free framework has permitted us to extend such formulas to make them applicable to nonsymmetric matrices. The extension of the unifying matrix formulas to nonsymmetric matrices is the subject matter of the present article. A conspicuous result is that in numerical experiments in 2D and 3D, the MF‐DP algorithms for nonsymmetric matrices exhibit an efficiency of the same order as state‐of‐the‐art algorithms for symmetric matrices, such as BDDC, FETI‐DP, and MF‐DP.© 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 27: 1262‐1289, 2011     

General theory of domain decomposition: Indirect methods

According to a general theory of domain decomposition methods (DDM), recently proposed by Herrera, DDM may be classified into two broad categories: direct and indirect (or Trefftz‐Herrera methods). This article is devoted to formulate systematically indirect methods and apply them to differential equations in several dimensions. They have interest since they subsume some of the best‐known formulations of domain decomposition methods, such as those based on the application of Steklov‐Poincaré operators. Trefftz‐Herrera approach is based on a special kind of Green's formulas applicable to discontinuous functions, and one of their essential features is the use of weighting functions which yield information, about the sought solution, at the internal boundary of the domain decomposition exclusively. A special class of Sobolev spaces is introduced in which boundary value problems with prescribed jumps at the internal boundary are formulated. Green's formulas applicable in such Sobolev spaces, which contain discontinuous functions, are established and from them the general framework for indirect methods is derived. Guidelines for the construction of the special kind of test functions are then supplied and, as an illustration, the method is applied to elliptic problems in several dimensions. A nonstandard method of collocation is derived in this manner, which possesses significant advantages over more standard procedures. © 2002 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 18: 296–322, 2002; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/num.10008     

Trefftz method: A general theory

A precise definition of Trefftz method is proposed and, starting with it, a general theory is briefly explained. This leads to formulating numerical methods from a domain decomposition perspective. An important feature of this approach is the systematic use of “fully discontinuous functions” and the treatment of a general boundary value problem with prescribed jumps. Usually finite element methods are developed using splines, but a more general point of view is obtained when they are formulated in spaces in which the functions together with their derivatives may have jump discontinuities and in the general context of boundary value problems with prescribed jumps. Two broad classes of Trefftz methods are obtained: direct (Trefftz—Jirousek) and indirect (Trefftz—Herrera) methods. In turn, each one of them can be divided into overlapping and nonoverlapping. The generality of the resulting theory is remarkable, because it is applicable to any partial (or ordinary) differential equation or system of such equations, which is linear. The article is dedicated to Professor Jiroslav Jirousek, who has been a very important driving force in the modern development of Trefftz method. © 2000 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 16: 561–580, 2000     

Collocation Trefftz methods for the stokes equations with singularity

This study explores the boundary methods for the two‐dimensional homogeneous Stokes equations and investigates the particular solutions (PS) satisfying the Stokes equations. Smooth solutions for the Stokes equations are provided by explicit fundamental solutions (FS) and PS in this study, and singular corner solutions are also provided from linear elastostatics given in Li et al. ([Eng. Anal. Bound. Elem. 34 (2009), 533‐648, 2009). A new singularity model with an interior crack is proposed and solved by the collocation Trefftz method (CTM). The proposed method achieves highly accurate solutions with the first leading coefficient having 10 significant digits. These solutions may be used as a benchmark for testing results obtained by other numerical methods. Error bounds are derived for the CTM solutions using the PS. For a general corner, the exponent ν_k in r can only be obtained by numerical solutions of a system of nonlinear algebraic equations. Therefore, the combined method using many FS plus a few singular solutions is inevitable in most applications. For singularity problems, combining a few singular solutions with the FS is an advanced topic and is successfully implemented in Lee et al. (Eng. Anal. Bound. Elem. 24 (2010), 632–654); however, combining a few singular solutions with the smooth PS fails to converge in the first leading coefficient. As a result, the aforementioned method is not applicable to the singularity problems addressed in this article. With the help of particular and singular solutions, the hybrid Trefftz method with Lagrange multipliers can be developed for the Stokes equations. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013     

Least‐square collocation and Lagrange multipliers forTaylor meshless method

A recently proposed meshless method is discussed in this article. It relies on Taylor series, the shape functions being high degree polynomials deduced from the Partial Differential Equation (PDE). In this framework, an efficient technique to couple several polynomial approximations has been presented in (Tampango, Potier‐Ferry, Koutsawa, Tiem, Int. J. Numer. Meth. Eng. vol. 95 (2013) pp. 1094–1112): the boundary conditions were applied using the least‐square collocation and the interface was coupled by a bridging technique based on Lagrange multipliers. In this article, least‐square collocation and Lagrange multipliers are applied for boundary conditions, respectively, and least‐square collocation is revisited to account for the interface conditions in piecewise resolutions. Various combinations of these two techniques have been investigated and the numerical results prove their effectiveness to obtain very accurate solutions, even for large scale problems.     

The multipliers‐free domain decomposition methods

This article concludes the development and summarizes a new approach to dual‐primal domain decomposition methods (DDM), generally referred to as “the multipliers‐free dual‐primal method.” Contrary to standard approaches, these new dual‐primal methods are formulated without recourse to Lagrange‐multipliers. In this manner, simple and unified matrix‐expressions, which include the most important dual‐primal methods that exist at present are obtained, which can be effectively applied to floating subdomains, as well. The derivation of such general matrix‐formulas is independent of the partial differential equations that originate them and of the number of dimensions of the problem. This yields robust and easy‐to‐construct computer codes. In particular, 2D codes can be easily transformed into 3D codes. The systematic use of the average and jump matrices, which are introduced in this approach as generalizations of the “average” and “jump” of a function, can be effectively applied not only at internal‐boundary‐nodes but also at edges and corners. Their use yields significant advantages because of their superior algebraic and computational properties. Furthermore, it is shown that some well‐known difficulties that occur when primal nodes are introduced are efficiently handled by the multipliers‐free dual‐primal method. The concept of the Steklov–Poincaré operator for matrices is revised by our theory and a new version of it, which has clear advantages over standard definitions, is given. Extensive numerical experiments that confirm the efficiency of the multipliers‐free dual‐primal methods are also reported here. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2010     

Local discontinuous galerkin approximation of convection‐dominated diffusion optimal control problems with control constraints

In this article, we investigate local discontinuous Galerkin approximation of stationary convection‐dominated diffusion optimal control problems with distributed control constraints. The state variable and adjoint state variable are approximated by piecewise linear polynomials without continuity requirement, whereas the control variable is discretized by variational discretization concept. The discrete first‐order optimality condition is derived. We show that optimization and discretization are commutative for the local discontinuous Galerkin approximation. Because the solutions to convection‐dominated diffusion equations often admit interior or boundary layers, residual type a posteriori error estimate in L² norm is proved, which can be used to guide mesh refinement. Finally, numerical examples are presented to illustrate the theoretical findings. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 339–360, 2014     

Boundary element tearing and interconnecting methods in unbounded domains

Finite element tearing and interconnecting (FETI) methods and boundary element tearing and interconnecting (BETI) methods are special iterative substructuring methods with Lagrange multipliers. For elliptic boundary value problems on bounded domains, the condition number of these methods can be rigorously bounded by C(1+log(H/h))², where H is the subdomain diameter and h the mesh size. The constant C is independent of H, h and possible jumps in the coefficients of the partial differential equation.In certain situations, e.g., in electromagnetic field computations, instead of imposing artificial boundary conditions one may be interested in modelling the real physical behaviour in an exterior domain with a radiation condition. In this work we analyze one-level BETI methods for such unbounded domains and show explicit condition number estimates similar to the one above. Our theoretical results are confirmed in numerical experiments.     

A priori error estimates for interior penalty versions of the local discontinuous Galerkin method applied to transport equations

The local discontinuous Galerkin method has been developed recently by Cockburn and Shu for convection‐dominated convection‐diffusion equations. In this article, we consider versions of this method with interior penalties for the numerical solution of transport equations, and derive a priori error estimates. We consider two interior penalty methods, one that penalizes jumps in the solution across interelement boundaries, and another that also penalizes jumps in the diffusive flux across such boundaries. For the first penalty method, we demonstrate convergence of order k in the L^∞(L²) norm when polynomials of minimal degree k are used, and for the second penalty method, we demonstrate convergence of order k+1/2. Through a parabolic lift argument, we show improved convergence of order k+1/2 (k+1) in the L²(L²) norm for the first penalty method with a penalty parameter of order one (h^?1). © 2001 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 17: 545–564, 2001     

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     

Convergence analysis of the LDG method applied to singularly perturbed problems

Considering a two‐dimensional singularly perturbed convection–diffusion problem with exponential boundary layers, we analyze the local discontinuous Galerkin (DG) method that uses piecewise bilinear polynomials on Shishkin mesh. A convergence rate O(N^‐1 lnN) in a DG‐norm is established under the regularity assumptions, while the total number of mesh points is O(N²). The rate of convergence is uniformly valid with respect to the singular perturbation parameter ε. Numerical experiments indicate that the theoretical error estimate is sharp. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2013     

Comparison principles applied to obstacle problems with penalties

Penalty methods form a well known technique to embed elliptic variational inequality problems into a family of variational equations (cf. [6], [13], [17]). Using the specific inverse monotonicity properties of these problems L _∞-bounds for the convergence can be derived by means of comparison solutions. Lagrange duality is applied to estimate parameters involved.

For piecewise linear finite elements applied on weakly acute triangulations in combination with mass lumping the inverse monotonicity of the obstacle problems can be transferred to its discretization. This forms the base of similar error estimations in the maximum norm for the penalty method applied to the discrete problem.

The technique of comparison solutions combined with the uniform boundedness of the Lagrange multipliers leads to decoupled convergence estimations with respect to the discretization and penalization parameters.     

Error analysis for hybrid Trefftz methods coupling traction conditions in linear elastostatics

For linear elastostatics, the Lagrange multiplier to couple the displacement (i.e., Dirichlet) condition is well known in mathematics community, but the Lagrange multiplier to couple the traction (i.e., Neumann) condition is popular for elasticity problems by the Trefftz method in engineering community, which is called the Hybrid Trefftz method (HTM). However, there has not been any analysis for these Lagrange multipliers to couple the traction condition so far. New error analysis of the HTM for elasticity problems is explored in this paper, to derive error bounds with the optimal convergence rates. Numerical experiments are reported to support this analysis. The error analysis of the HTM for linear elastostatics is the main aim of this paper. In this paper, the collocation Trefftz method (CTM) without a multiplier is also introduced, accompanied with error analysis. Numerical comparisons are made for HTM and CTM using fundamental solutions (FS) and particular solutions (PS). The error analysis and numerical computations show that the accuracy of the HTM is equivalent to that of the CTM, but the stability of the CTM is good. For elasticity and other complicated problems, the simplicity of algorithms and programming grants the CTM a remarkable advantage. More numerical comparisons show that using PS is more efficient than using FS in both HTM and CTM. However, since the optimal convergence rates are the most important criterion in evaluation of numerical methods, the global performance of the HTM is as good as that of the CTM. The comparisons of HTM and CTM using FS and PS are the next aim of this article. © 2011 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2011     

An Error Analysis of Discontinuous Finite Element Methods for the Optimal Control Problems Governed by Stokes Equation

In this article, an abstract framework for the error analysis of discontinuous finite element method is developed for the distributed and Neumann boundary control problems governed by the stationary Stokes equation with control constraints. A priori error estimates of optimal order are derived for velocity and pressure in the energy norm and the L²-norm, respectively. Moreover, a reliable and efficient a posteriori error estimator is derived. The results are applicable to a variety of problems just under the minimal regularity possessed by the well-posedness of the problem. In particular, we consider the abstract results with suitable stable pairs of velocity and pressure spaces like as the lowest-order Crouzeix–Raviart finite element and piecewise constant spaces, piecewise linear and constant finite element spaces. The theoretical results are illustrated by the numerical experiments.     

A posteriori error estimates for the mixed finite element method with Lagrange multipliers

We consider the mixed finite element method with Lagrange multipliers as applied to second‐order elliptic equations in divergence form with mixed boundary conditions. The corresponding Galerkin scheme is defined by using Raviart‐Thomas spaces. We develop a posteriori error analyses yielding a reliable and efficient estimate based on residuals, and a reliable and quasi‐efficient estimate based on local problems, respectively. Several numerical results illustrate the suitability of these a posteriori estimates for the adaptive computation of the discrete solutions. © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2005     

A special complementarity function revisited

ABSTRACT

Recently, a local framework of Newton-type methods for constrained systems of equations has been developed. Applied to the solution of Karush–Kuhn–Tucker (KKT) systems, the framework enables local quadratic convergence under conditions that allow nonisolated and degenerate KKT points. This result is based on a reformulation of the KKT conditions as a constrained piecewise smooth system of equations. It is an open question whether a comparable result can be achieved for other (not piecewise smooth) reformulations. It will be shown that this is possible if the KKT system is reformulated by means of the Fischer–Burmeister complementarity function under conditions that allow degenerate KKT points and nonisolated Lagrange multipliers. To this end, novel constrained Levenberg–Marquardt subproblems are introduced. They allow significantly longer steps for updating the multipliers. Based on this, a convergence rate of at least 1.5 is shown.     

A reliable treatment of singular Emden-Fowler initial value problems and boundary value problems

In this paper, the variational iteration method (VIM) is used to study the singular Emden-Fowler initial value problems and boundary value problems arising in physics and astrophysics. The VIM overcomes the singularity at the origin. The Lagrange multipliers for all cases of the equations are determined. The work is supported by analyzing few initial value problems and boundary value problems where the convergence of the results is emphasized.     

A uniform optimal‐order estimate for an Eulerian‐Lagrangian discontinuous Galerkin method for transient advection–diffusion equations

We prove an optimal‐order error estimate in a weighted energy norm for the Eulerian‐Lagrangian discontinuous Galerkin method for unsteady‐state advection–diffusion equations with general inflow and outflow boundary conditions. It is well‐known that these problems admit dynamic fronts with interior and boundary layers. The estimate holds uniformly with respect to the vanishing diffusion coefficient. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009     

The Singular Function Boundary Integral Method for singular Laplacian problems over circular sections

The Singular Function Boundary Integral Method (SFBIM) for solving two-dimensional elliptic problems with boundary singularities is revisited. In this method the solution is approximated by the leading terms of the asymptotic expansion of the local solution, which are also used to weight the governing partial differential equation. The singular coefficients, i.e., the coefficients of the local asymptotic expansion, are thus primary unknowns. By means of the divergence theorem, the discretized equations are reduced to boundary integrals and integration is needed only far from the singularity. The Dirichlet boundary conditions are then weakly enforced by means of Lagrange multipliers, the discrete values of which are additional unknowns. In the case of two-dimensional Laplacian problems, the SFBIM converges exponentially with respect to the numbers of singular functions and Lagrange multipliers. In the present work the method is applied to Laplacian test problems over circular sectors, the analytical solution of which is known. The convergence of the method is studied for various values of the order p of the polynomial approximation of the Lagrange multipliers (i.e., constant, linear, quadratic, and cubic), and the exact approximation errors are calculated. These are compared to the theoretical results provided in the literature and their agreement is demonstrated.