Showing papers in "Mathematics of Computation in 2000"
TL;DR: A number of new local and parallel discretization and adaptive nite element algorithms are proposed and analyzed in this paper for elliptic boundary value problems and the main idea is to use a coarse grid to approximate the low frequencies and then to correct the resulted residue (which contains mostly high frequencies) by some local/parallel procedures.
Abstract: A number of new local and parallel discretization and adaptive nite element algorithms are proposed and analyzed in this paper for elliptic boundary value problems. These algorithms are motivated by the observation that, for a solution to some elliptic problems, low frequency components can be approximated well by a relatively coarse grid and high frequency components can be computed on a ne grid by some local and parallel procedure. The theoretical tools for analyzing these methods are some local a priori and a posteriori estimates that are also obtained in this paper for nite element solutions on general shape-regular grids. Some numerical experiments are also presented to support the theory. In this paper, we will propose some new parallel techniques for nite element computation. These techniques are based on our understanding of the local and global properties of a nite element solution to some elliptic problems. Simply speaking, the global behavior of a solution is mostly governed by low frequency components while the local behavior is mostly governed by high frequency compo- nents. The main idea of our new algorithms is to use a coarse grid to approximate the low frequencies and then to use a ne grid to correct the resulted residue (which contains mostly high frequencies) by some local/parallel procedures. Let us now give a somewhat more detailed but informal (and hopefully infor- mative) description of the main ideas and results in this paper. We consider the following very simple model problem posed on a convex polygonal domain R 2 : ( u + bru = f; in ;
209 citations
TL;DR: It is shown that the main classes of a posteriori error estimates known in the literature follow from the duality error estimate obtained and, thus, can be justified via theDuality theory.
Abstract: The objective of this paper is to introduce a general scheme for deriving a posteriori error estimates by using duality theory of the calculus of variations. We consider variational problems of the form inf {F(v) + G(Λv)}, where F: V → R is a convex lower semicontinuous functional, G: Y → R is a uniformly convex functional, V and Y are reflexive Banach spaces, and Λ: V → Y is a bounded linear operator. We show that the main classes of a posteriori error estimates known in the literature follow from the duality error estimate obtained and, thus, can be justified via the duality theory.
191 citations
TL;DR: A general approach for connecting covolume or finite volume element methods for variable coefficient elliptic and parabolic problems on convex smooth domains in the plane is introduced and the maximur norm error in the gradient is shown to be of first order.
Abstract: In this paper we consider covolume or finite volume element methods for variable coefficient elliptic and parabolic problems on convex smooth domains in the plane. We introduce a general approach for connecting these methods with finite element method analysis. This unified approach is used to prove known convergence results in the H 1 , L 2 norms and new results in the max-norm. For the elliptic problems we demonstrate that the error u-u h between the exact solution u and the approximate solution u h in the maximum norm is O(h 2 |ln h|) in the linear element case. Furthermore, the maximur norm error in the gradient is shown to be of first order. Similar results hold for the parabolic problems.
184 citations
TL;DR: This paper presents examples of mixed finite element approximations that satisfy the above properties but exhibit spurious eigenvalues, and proves that bad behavior is proved analytically and demonstrated in numerical experiments.
Abstract: In the approximation of linear elliptic operators in mixed form, it is well known that the so-called inf-sup and ellipticity in the kernel properties are sufficient (and, in a sense to be made precise, necessary) in order to have good approximation properties and optimal error bounds. One might think, in the spirit of Mercier-Osborn-Rappaz-Raviart and in consideration of the good behavior of commonly used mixed elements (like Raviart-Thomas or Brezzi-Douglas-Marini elements), that these conditions are also sufficient to ensure good convergence properties for eigenvalues. In this paper we show that this is not the case. In particular we present examples of mixed finite element approximations that satisfy the above properties but exhibit spurious eigenvalues. Such bad behavior is proved analytically and demonstrated in numerical experiments. We also present additional assumptions (fulfilled by the commonly used mixed methods already mentioned) which guarantee optimal error bounds for eigenvalue approximations as well.
182 citations
TL;DR: This work shows how to apply a Pollard lambda search on a set of equivalence classes derived from E, which requires fewer iterations than the standard approach and speeds up the standard algorithm by a factor of √2m.
Abstract: The best algorithm known for finding logarithms on an elliptic curve (E) is the (parallelized) Pollard lambda collision search. We show how to apply a Pollard lambda search on a set of equivalence classes derived from E, which requires fewer iterations than the standard approach. In the case of anomalous binary curves over F 2 m, the new approach speeds up the standard algorithm by a factor of √2m.
134 citations
TL;DR: These tables record results on curves with many points over finite fields by giving in two tables the best presently known bounds for N q (g), the maximum number of rational points on a smooth absolutely irreducible projective curve of genus g over a field F q of cardinality q.
Abstract: These tables record results on curves with many points over finite fields. For relatively small genus (0 < g < 50) and q a small power of 2 or 3 we give in two tables the best presently known bounds for N q (g), the maximum number of rational points on a smooth absolutely irreducible projective curve of genus g over a field F q of cardinality q. In additional tables we list for a given pair (g, q) the type of construction of the best curve so far, and we give a reference to the literature where such a curve can be found.
132 citations
TL;DR: This paper constructs elliptic boundary value problems whose standard finite element approximations converge arbitrarily slowly in the energy norm, and shows that adaptive procedures cannot improve this slow convergence.
Abstract: In this paper we construct elliptic boundary value problems whose standard finite element approximations converge arbitrarily slowly in the energy norm, and show that adaptive procedures cannot improve this slow convergence. We also show that the L 2 -norm and the nodal point errors converge arbitrarily slowly. With the L 2 -norm two cases need to be distinguished, and the usual duality principle does not characterize the error completely. The constructed elliptic problems are one dimensional.
124 citations
TL;DR: A posteriori error estimates in the L 1 -norm for upwind finite volume schemes for the discretization of nonlinear conservation laws on unstructured grids in multi dimensions are derived.
Abstract: In this paper we shall derive a posteriori error estimates in the L 1 -norm for upwind finite volume schemes for the discretization of nonlinear conservation laws on unstructured grids in multi dimensions. This result is mainly based on some fundamental a priori error estimates published in a recent paper by C. Chainais-Hillairet. The theoretical results are confirmed by numerical experiments.
119 citations
TL;DR: A Lanczos-type algorithm that extends the classical Lanczos process for single starting vectors to mul- tiple starting vectors and can handle the most general case of right and left start- ing blocks of arbitrary sizes.
Abstract: Given a square matrix and single right and left starting vectors, the classical nonsymmetric Lanczos process generates two sequences of bior- thogonal basis vectors for the right and left Krylov subspaces induced by the given matrix and vectors. In this paper, we propose a Lanczos-type algorithm that extends the classical Lanczos process for single starting vectors to mul- tiple starting vectors. Given a square matrix and two blocks of right and left starting vectors, the algorithm generates two sequences of biorthogonal basis vectors for the right and left block Krylov subspaces induced by the given data. The algorithm can handle the most general case of right and left start- ing blocks of arbitrary sizes, while all previously proposed extensions of the Lanczos process are restricted to right and left starting blocks of identical sizes. Other features of our algorithm include a built-in deation procedure to detect and delete linearly dependent vectors in the block Krylov sequences, and the option to employ look-ahead to remedy the potential breakdowns that may occur in nonsymmetric Lanczos-type methods.
114 citations
TL;DR: Two explicit error representation formulas are derived for degenerate parabolic PDEs, which are based on evaluating a parabolic residual in negative norms, and used in designing an efficient adaptive algorithm, which equidistributes space and time discretization errors via refinement/coarsening.
Abstract: Two explicit error representation formulas are derived for degenerate parabolic PDEs, which are based on evaluating a parabolic residual in negative norms. The resulting upper bounds are valid for any numerical method, and rely on regularity properties of solutions of a dual parabolic problem in nondivergence form with vanishing diffusion coefficient. They are applied to a practical space-time discretization consisting of C 0 piecewise linear finite elements over highly graded unstructured meshes, and backward finite differences with varying time-steps. Two rigorous a posteriori error estimates are derived for this scheme, and used in designing an efficient adaptive algorithm, which equidistributes space and time discretization errors via refinement/coarsening. A simulation finally compares the behavior of the rigorous a posteriori error estimators with a heuristic approach, and hints at the potentials and reliability of the proposed method.
107 citations
TL;DR: A new algorithm is described that does not require the entries of the tridiagonal matrix to be determined, and thereby avoids computations that can be sensitive to perturbations.
Abstract: Recently Laurie presented a new algorithm for the computation of (2n+1)-point Gauss-Kronrod quadrature rules with real nodes and positive weights. This algorithm first determines a symmetric tridiagonal matrix of order 2n + 1 from certain mixed moments, and then computes a partial spectral factorization. We describe a new algorithm that does not require the entries of the tridiagonal matrix to be determined, and thereby avoids computations that can be sensitive to perturbations. Our algorithm uses the consolidation phase of a divide-and-conquer algorithm for the symmetric tridiagonal eigenproblem. We also discuss how the algorithm can be applied to compute Kronrod extensions of Gauss-Radau and Gauss-Lobatto quadrature rules. Throughout the paper we emphasize how the structure of the algorithm makes efficient implementation on parallel computers possible. Numerical examples illustrate the performance of the algorithm.
TL;DR: This paper considers iterative algorithms of Uzawa type for solving linear nonsymmetric saddle point problems, and considers systems, written as usual in block form, where the upper left block is an invertible linear operator with positive definite symmetric part.
Abstract: In this paper, we consider iterative algorithms of Uzawa type for solving linear nonsymmetric saddle point problems. Specifically, we consider systems, written as usual in block form, where the upper left block is an invertible linear operator with positive definite symmetric part. Such saddle point problems arise, for example, in certain finite element and finite difference discretizations of Navier-Stokes equations, Oseen equations, and mixed finite element discretization of second order convection-diffusion problems. We consider two algorithms, each of which utilizes a preconditioner for the operator in the upper left block. Convergence results for the algorithms are established in appropriate norms. The convergence of one of the algorithms is shown assuming only that the preconditioner is spectrally equivalent to the inverse of the symmetric part of the operator. The other algorithm is shown to converge provided that the preconditioner is a sufficiently accurate approximation of the inverse of the upper left block. Applications to the solution of steady-state Navier-Stokes equations are discussed, and, finally, the results of numerical experiments involving the algorithms are presented.
TL;DR: The time discretization of an initial-value problem for a homogeneous abstract parabolic equation is treated by first using a representation of the solution as an integral along the boundary of a sector in the right half of the complex plane, then transforming this into a real integral on the finite interval, and finally applying a standard quadrature formula to this integral.
Abstract: We treat the time discretization of an initial-value problem for a homogeneous abstract parabolic equation by first using a representation of the solution as an integral along the boundary of a sector in the right half of the complex plane, then transforming this into a real integral on the finite interval , and finally applying a standard quadrature formula to this integral. The method requires the solution of a finite set of elliptic problems with complex coefficients, which are independent and may therefore be done in parallel. The method is combined with spatial discretization by finite elements.
TL;DR: The main result is that finding the number 1.13198824 involves the theory of random matrix products, Stern-Brocot division of the real line, a fractal-like measure, a computer calculation, and a rounding error analysis to validate the computer calculation.
Abstract: \begin{} For the familiar Fibonacci sequence --defined by $f_1 = f_2 = 1$, and $f_n = f_{n-1} + f_{n-2}$ for $n<2$ --$f_n$ increases exponentially with $n$ at a rate given by the golden ratio $(1+\sqrt{5})/2=1.61803398\ldots$. But for a simple modification with both additions and subtractions --the {\it random} Fibonacci sequences defined by $t_1=t_2=1$, and for $n<2$, $t_n = \pm t_{n-1} \pm t_{n-2}$, where each $\pm$ sign is independent and either $+$ or $-$ with probability $1/2$ --it is not even obvious if $\abs{t_n}$ should increase with $n$. Our main result is that \begin{equation*} \sqrt[n]{\abs{t_n}} \rightarrow 1.13198824\ldots\:\:\: \text{as}\:\:\: n \rightarrow\infty \end{equation*} with probability $1$. Finding the number $1.13198824\ldots$ involves the theory of random matrix products, Stern-Brocot division of the real line, a fractal-like measure, a computer calculation, and a rounding error analysis to validate the computer calculation. \end{abstract}
TL;DR: Newton's method is studied for overdetermined systems where there are more equations than unknowns and Newton's method for such a system is studied.
Abstract: Complexity theoretic aspects of continuation methods for the solution of square or underdetermined systems of polynomial equations have been studied by various authors. In this paper we consider overdetermined systems where there are more equations than unknowns. We study Newton's method for such a system.
TL;DR: A finite-difference scheme for the approximation of partial differential equations in R 1, with additional stochastic noise, using the weighted Lp-theory of SPDE and a sup-norm error estimate is derived and the rate of convergence is given.
Abstract: The paper concerns finite-difference scheme for the approximation of partial differential equations in R 1 , with additional stochastic noise. By replacing the space derivatives in the original stochastic partial differential equation (SPDE, for short) with difference quotients, we obtain a system of stochastic ordinary differential equations. We study the difference between the solution of the original SPDE and the solution to the corresponding equation obtained by discretizing the space variable. The need to approximate the solution in R 1 with functions of compact support requires us to introduce a scale of weighted Sobolev spaces. Employing the weighted Lp-theory of SPDE, a sup-norm error estimate is derived and the rate of convergence is given.
TL;DR: This paper provides a framework for developing computationally efficient multilevel preconditioners and representations for Sobolev norms by constructing operators which are spectrally equivalent to those of the form A = Σ k μ k (Q k - Q k-1 ).
Abstract: This paper provides a framework for developing computationally efficient multilevel preconditioners and representations for Sobolev norms. Specifically, given a Hilbert space V and a nested sequence of subspaces V 1 C V 2 C... C V, we construct operators which are spectrally equivalent to those of the form A = Σ k μ k (Q k - Q k-1 ). Here μ k , k=1,2,..., are positive numbers and Q k is the orthogonal projector onto V k with Q 0 = 0. We first present abstract results which show when A is spectrally equivalent to a similarly constructed operator A defined in terms of an approximation Q k of Q k , for k = 1,2,... We show that these results lead to efficient preconditioners for discretizations of differential and pscudo-differential operators of positive and negative order. These results extend to sums of operators. For example, singularly perturbed problems such as I -eΔ can be preconditioned uniformly independently of the parameter e. We also show how to precondition an operator which results from Tikhonov regularization of a problem with noisy data. Finally, we describe how the technique provides computationally efficient bounded discrete extensions which have applications to domain decomposition.
TL;DR: Some ideas based on the use of the Strang splitting for the approximation of matrix exponentials are presented, in tandem with general theory.
Abstract: Consider a differential equation y' = A(t,y)y, y(0) = y0 with y 0 ∈ G and A: R + × G → g, where g is a Lie algebra of the matricial Lie group G. Every B ∈ g can be mapped to G by the matrix exponential map exp (tB) with t ∈ R. Most numerical methods for solving ordinary differential equations (ODEs) on Lie groups are based on the idea of representing the approximation y n of the exact solution y(t n ), t n ∈ R + , by means of exact exponentials of suitable elements of the Lie algebra, applied to the initial value y 0 . This ensures that .y n ∈ G. When the exponential is difficult to compute exactly, as is the case when the dimension is large, an approximation of exp (tB) plays an important role in the numerical solution of ODEs on Lie groups. In some cases rational or polynomial approximants are unsuitable and we consider alternative techniques, whereby exp (tB) is approximated by a product of simpler exponentials. In this paper we present some ideas based on the use of the Strang splitting for the approximation of matrix exponentials. Several cases of g and G are considered, in tandem with general theory. Order conditions are discussed, and a number of numerical experiments conclude the paper.
TL;DR: It is shown that a further modified Hotta-Yoshise algorithm is globally and superlinearly convergent, with a convergence Q-order 1 + t, under suitable conditions, where t ∈ (0, 1) is an additional parameter.
Abstract: Recently, based upon the Chen-Harker-Kanzow-Smale smoothing function and the trajectory and the neighbourhood techniques, Hotta and Yoshise proposed a noninterior point algorithm for solving the nonlinear complementarity problem. Their algorithm is globally convergent under a relatively mild condition. In this paper, we modify their algorithm and combine it with the superlinear convergence theory for nonlinear equations. We provide a globally linearly convergent result for a slightly updated version of the Hotta-Yoshise algorithm and show that a further modified Hotta-Yoshise algorithm is globally and superlinearly convergent, with a convergence Q-order 1 + t, under suitable conditions, where t ∈ (0, 1) is an additional parameter.
TL;DR: With the approach presented, much simpler data access patterns and similar or lower computational counts can be achieved through propagating a family of univariate Taylor series of a suitable degree.
Abstract: This article considers the problem of evaluating all pure and mixed partial derivatives of some vector function defined by an evaluation procedure. The natural approach to evaluating derivative tensors might appear to be their recursive calculation in the usual forward mode of computational differentiation. However, with the approach presented in this article, much simpler data access patterns and similar or lower computational counts can be achieved through propagating a family of univariate Taylor series of a suitable degree. It is applicable for arbitrary orders of derivatives. Also it is possible to calculate derivatives only in some directions instead of the full derivative tensor. Explicit formulas for all tensor entries as well as estimates for the corresponding computational complexities are given.
TL;DR: The ultraconvergence property of a gradient recovery technique proposed by Zienkiewicz and Zhu is analyzed for the Laplace equation in the two dimensional setting and it is shown that the convergence rate of the recovered gradient at an interior node is two orders higher than the optimal global convergence rate.
Abstract: The ultraconvergence property of a gradient recovery technique proposed by Zienkiewicz and Zhu is analyzed for the Laplace equation in the two dimensional setting. Under the assumption that the pollution effect is not present or is properly controlled, it is shown that the convergence rate of the recovered gradient at an interior node is two orders higher than the optimal global convergence rate when even-order finite element spaces and local uniform rectangular meshes are used.
TL;DR: An additive, two--level overlapping Schwarz preconditioner is proposed, consisting of a coarse problem on a coarse triangulation and local solvers associated to suitable problems defined on a family of subdomains, which is a generalization of the corresponding overlapping method for approximations on continuous finite element spaces.
Abstract: We consider a scalar advection-diffusion problem and a recently proposed discontinuous Galerkin approximation, which employs discontinuous finite element spaces and suitable bilinear forms containing interface terms that ensure consistency. For the corresponding sparse, non-symmetric linear system, we propose and study an additive, two--level overlapping Schwarz preconditioner, consisting of a coarse problem on a coarse triangulation and local solvers associated to suitable problems defined on a family of subdomains. This is a generalization of the corresponding overlapping method for approximations on continuous finite element spaces. Related to the lack of continuity of our approximation spaces, some interesting new features arise in our generalization, which have no analog in the conforming case. We prove an upper bound for the number of iterations obtained by using this preconditioner with GMRES, which is independent of the number of degrees of freedom of the original problem and the number of subdomains. The performance of the method is illustrated by several numerical experiments for different test problems, using linear finite elements in two dimensions.
TL;DR: The convergence of a class of finite-differences numerical schemes is studied and an appropriate concept of consistency with the continuous problem is introduced.
Abstract: Conservative linear equations arise in many areas of application, including continuum mechanics or high-frequency geometrical optics approximations. This kind of equations admits most of the time solutions which are only bounded measures in the space variable known as duality solutions. In this paper, we study the convergence of a class of finite-differences numerical schemes and introduce an appropriate concept of consistency with the continuous problem. Some basic examples including computational results are also supplied.
TL;DR: An a posteriori stopping rule is suggested to choose the stopping index of iteration, and with the integer k δ determined by this rule it is proved that the convergence of x kδ δ is obtained, and the rate of convergence is derived when x 0 - x satisfies a suitable source-wise representation.
Abstract: The iteratively regularized Gauss-Newton method is applied to compute the stable solutions to nonlinear ill-posed problems F(x) = y when the data y is given approximately by y δ with ∥y δ - y∥ ≤ δ. In this method, the iterative sequence {x k δ } is defined successively by x k +1 δ = x k δ -(α k I + F' (x k δ ) * F' (x k δ )) -1 (F'(x k δ ) * (F(x k δ )-y δ ) + α k (x k δ - x 0 )), where x 0 δ := x 0 is an initial guess of the exact solution x and {α k } is a given decreasing sequence of positive numbers admitting suitable properties. When x k δ is used to approximate x , the stopping index should be designated properly. In this paper, an a posteriori stopping rule is suggested to choose the stopping index of iteration, and with the integer k δ determined by this rule it is proved that ∥x k δ - x∥ ≤ C inf {∥x k - x ∥ + δ/√αk∥: k = 0,1,...} with a constant C independent of δ, where x k denotes the iterative solution corresponding to the noise free case. As a consequence of this result, the convergence of x kδ δ is obtained, and moreover the rate of convergence is derived when x 0 - x satisfies a suitable source-wise representation. The results of this paper suggest that the iteratively regularized Gauss-Newton method, combined with our stopping rule, defines a regularization method of optimal order for each 0 < v < 1. Numerical examples for parameter estimation of a differential equation are given to test the theoretical results.
TL;DR: The number field sieve factoring algorithm is conjectured to factor a number the size of q in the same amount of time when restricted to finite fields of an arbitrary but fixed degree.
Abstract: We describe an adaptation of the number field sieve to the problem of computing logarithms in a finite field. We conjecture that the running time of the algorithm, when restricted to finite fields of an arbitrary but fixed degree, is L q [1/3; (64/9) 1/3 + o(1)], where q is the cardinality of the field, L q [s; c] = exp(c(log q) s (log log q) 1- s ), and the o(1) is for q → ∞. The number field sieve factoring algorithm is conjectured to factor a number the size of q in the same amount of time.
TL;DR: The main idea of the evolution Galerkin methods is the following: the initial function is evolved using the characteristic cone and then projected onto a finite element space.
Abstract: , The subject of the paper is the analysis of three new evolution Galerkin schemes for a system of hyperbolic equations, and particularly for the wave equation system. The aim is to construct methods which take into account all of the infinitely many directions of propagation of bicharacteristics. The main idea of the evolution Galerkin methods is the following: the initial function is evolved using the characteristic cone and then projected onto a finite element space. A numerical comparison is given of the new methods with already existing methods, both those based on the use of bicharacteristics as well as commonly used finite difference and finite volume methods. We discuss the stability properties of the schemes and derive error estimates.
TL;DR: A theoretical analysis of the proposed gauge formulation of the incompressible Navier-Stokes equations in terms of an auxiliary field a and a gauge variable Φ, u = a + ⊇Φ and it is proved first order convergence of the gauge method when the authors use MAC grids as their spatial discretization.
Abstract: A new formulation, a gauge formulation of the incompressible Navier-Stokes equations in terms of an auxiliary field a and a gauge variable Φ, u = a + ⊇Φ, was proposed recently by E and Liu. This paper provides a theoretical analysis of their formulation and verifies the computational advantages. We discuss the implicit gauge method, which uses backward Euler or Crank-Nicolson in time discretization. However, the boundary conditions for the auxiliary field a are implemented explicitly (vertical extrapolation). The resulting momentum equation is decoupled from the kinematic equation, and the computational cost is reduced to solving a standard heat and Poisson equation. Moreover, such explicit boundary conditions for the auxiliary field a will be shown to be unconditionally stable for Stokes equations. For the full nonlinear Navier-Stokes equations the time stepping constraint is reduced to the standard CFL constraint Δt/Δx < C. We also prove first order convergence of the gauge method when we use MAC grids as our spatial discretization. The optimal error estimate for the velocity field is also obtained.
TL;DR: The mortar finite element is an example of a non-conforming method which can be used to decompose and re-compose a domain into sub-domains without requiring compatibility between the meshes on the separate components.
Abstract: The mortar finite element is an example of a non-conforming method which can be used to decompose and re-compose a domain into sub-domains without requiring compatibility between the meshes on the separate components. We obtain stability and convergence results for this method that are uniform in terms of both the degree and the mesh used, without assuming quasiuniformity for the meshes. Our results establish that the method is optimal when non-quasiuniform h or hp methods are used. Such methods are essential in practice for good rates of convergence when the interface passes through a corner of the domain. We also give an error estimate for when the p version is used. Numerical results for hp and hp mortar FEMs show that these methods behave as well as conforming FEMs. An hp extension theorem is also proved.
TL;DR: The main results are that a new optimal a priori L 2 error estimate of a least-squares mixed finite element method for a steady convection-diffusion problem is developed and that four fully-discrete least-Squares Mixed finite element schemes for an initial-boundary value problem of a nonlinear nonstationary convection -diffusion equation are formulated.
Abstract: Some least-squares mixed finite element methods for convection-diffusion problems, steady or nonstationary, are formulated, and convergence of these schemes is analyzed. The main results are that a new optimal a priori L 2 error estimate of a least-squares mixed finite element method for a steady convection-diffusion problem is developed and that four fully-discrete least-squares mixed finite element schemes for an initial-boundary value problem of a nonlinear nonstationary convection-diffusion equation are formulated. Also, some systematic theories on convergence of these schemes are established.
TL;DR: This analysis generalizes the results of Monk by means of the technique of integral identity on a rectangular mesh and generalized into more general domains and problems with the variable coefficients.
Abstract: In this paper, the global superconvergence is analysed on two schemes (a mixed finite element scheme and a finite element scheme) for Maxwell's equations in R 3 , Such a supercovergence analysis is achieved by means of the technique of integral identity (which has been used in the supercovergence analysis for many other equations and schemes) on a rectangular mesh, and then are generalized into more general domains and problems with the variable coefficients. Besides being more direct, our analysis generalizes the results of Monk.