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We consider a class of “filtered” schemes for first order time dependent Hamilton–Jacobi equations and prove a general convergence result for this class of schemes. A typical filtered scheme is obtained mixing a high-order scheme and a monotone scheme according to a filter function F which decides where the scheme has to switch from one scheme to the other. A crucial role for this switch is played by a parameter $$\varepsilon =\varepsilon ({\Delta t,\Delta x})>0$$ which goes to 0 as the time and space steps $$(\Delta t,\Delta x)$$ are going to 0 and does not depend on the time $$t_n$$, for each iteration n. The tuning of this parameter in the code is rather delicate and has an influence on the global accuracy of the filtered scheme. Here we introduce an adaptive and automatic choice of $$\varepsilon =\varepsilon ^n (\Delta t, \Delta x)$$ at every iteration modifying the classical set up. The adaptivity is controlled by a smoothness indicator which selects the regions where we modify the regularity threshold $$\varepsilon ^n$$. A convergence result and some error estimates for the new adaptive filtered scheme are proved, this analysis relies on the properties of the scheme and of the smoothness indicators. Finally, we present some numerical tests to compare the adaptive filtered scheme with other methods.

This paper is concerned with monotone (time-explicit) finite difference schemes associated with first order Hamilton-Jacobi equations posed on a junction. They extend the schemes recently introduced by Costeseque, Lebacque and Monneau (2013) to general junction conditions. On the one hand, we prove the convergence of the numerical solution towards the viscosity solution of the Hamilton-Jacobi equation as the mesh size tends to zero for general junction conditions. On the other hand, we derive optimal error estimates of order $(\Delta x)^{\frac{1}{2}}$ in $L_{loc}^{\infty}$ for junction conditions of optimal-control type at least if the flux is "strictly limited".

We consider optimal control problems of quasilinear elliptic equations with gradient coefficients arising in variable viscosity fluid flow. The state equation is monotone and the controls are of distributed type. We prove that the control-to-state operator is twice Frechet differentiable for this class of equations. A finite element approximation is studied and an estimate of optimal order h is obtained for the control. The result makes use of the distributed structure of the controls, together with a regularity estimate for elliptic equations with Holder coefficients and a second order sufficient optimality condition. The paper ends with a numerical experiment, where the approximation order is computationally tested.

Continuously operated clarifier–thickener (CT) units can be modeled by a non-linear, scalar conservation law with a flux that involves two parameters that depend discontinuously on the space variable. This paper presents two numerical schemes for the solution of this equation that have formal second-order accuracy in both the time and space variable. One of the schemes is based on standard total variation diminishing (TVD) methods, and is addressed as a simple TVD (STVD) scheme, while the other scheme, the so-called flux-TVD (FTVD) scheme, is based on the property that due to the presence of the discontinuous parameters, the flux of the solution (rather than the solution itself) has the TVD property. The FTVD property is enforced by a new nonlocal limiter algorithm. We prove that the FTVD scheme converges to a BV t solution of the conservation law with discontinuous flux. Numerical examples for both resulting schemes are presented. They produce comparable numerical errors, while the FTVD scheme is supported by convergence analysis. The accuracy of both schemes is superior to that of the monotone first-order scheme based on the adaptation of the Engquist–Osher scheme to the discontinuous flux setting of the CT model (Burger, Karlsen and Towers in SIAM J Appl Math 65:882–940, 2005). In the CT application there is interest in modelling sediment compressibility by an additional strongly degenerate diffusion term. Second-order schemes for this extended equation are obtained by combining either the STVD or the FTVD scheme with a Crank–Nicolson discretization of the degenerate diffusion term in a Strang-type operator splitting procedure. Numerical examples illustrate the resulting schemes.

We consider a singularly perturbed one-dimensional reaction–diffusion problem with strong layers. The problem is discretized using a compact fourth order finite difference scheme. Altough the discretization is not inverse monotone we are able to establish its maximum-norm stability and to prove its pointwise convergence on a Shishkin mesh. The convergence is uniform with respect to the perturbation parameter. Numerical experiments complement our theoretical results.

A successive relaxation iterative algorithm for discrete HJB equations is proposed. Monotone convergence has been proved for the algorithm.

We present a local exponential fitting hybridized mixed finite-element method for convection–diffusion problem on a bounded domain with mixed Dirichlet Neuman boundary conditions. With a new technique that interpretes the algebraic system after static condensation as a bilinear form acting on certain lifting operators we prove an a priori error estimate on the Lagrange multipliers that requires minimal regularity. While an extension of more classical arguments provide an estimate for the other solution components.

Non-oscillatory schemes are widely used in numerical approximations of nonlinear conservation laws. The Nessyahu–Tadmor (NT) scheme is an example of a second order scheme that is both robust and simple. In this paper, we prove a new stability property of the NT scheme based on the standard minmod reconstruction in the case of a scalar strictly convex conservation law. This property is similar to the One-sided Lipschitz condition for first order schemes. Using this new stability, we derive the convergence of the NT scheme to the exact entropy solution without imposing any nonhomogeneous limitations on the method. We also derive an error estimate for monotone initial data.

We study dislocation dynamics with a level set point of view. The model we present here looks at the zero level set of the solution of a non local Hamilton Jacobi equation, as a dislocation in a plane of a crystal. The front has a normal speed, depending on the solution itself. We prove existence and uniqueness for short time in the set of continuous viscosity solutions. We also present a first order finite difference scheme for the corresponding level set formulation of the model. The scheme is based on monotone numerical Hamiltonian, proposed by Osher and Sethian. The non local character of the problem makes it not monotone. We obtain an explicit convergence rate of the approximate solution to the viscosity solution. We finally provide numerical simulations.

This paper is concerned with finite difference solutions of a system of reaction-diffusion equations with coupled nonlinear boundary conditions and time delays. The reaction functions and the boundary functions are not necessarily quasimonotone, and the time delays may appear in the reaction functions as well as in the boundary functions. The investigation is devoted to the finite difference system for both the time-dependent problem and its corresponding steady-state problem. Some monotone iteration processes for the finite difference systems are given, and the asymptotic behavior of the time-dependent solution in relation to the steady-state solution is discussed. The asymptotic behavior result leads to some local and global attractors of the time-dependent problem, including the convergence of the time-dependent solution to a unique steady-state solution. An application and some numerical results to an enzyme-substrate reaction-diffusion problem are given. All the results are directly applicable to parabolic-ordinary systems and to reaction-diffusion systems without time delays.

In [2] Bermudez and Moreno introduced a duality algorithm for the numerical solution of variational inequalities; this algorithm is based on some properties of the Yosida regularization of maximal monotone operators. The performances of this algorithm strongly depend on the choice of two constant parameters. A generalization of the algorithm with automatic choice of parameters was discussed in [13], where the constant parameters were replaced by scalar functions, thus improving the convergence of the algorithm. In this article we present a generalization of the Bermudez-Moreno algorithm that allows the use of very general operators as parameters, extending some of the results in [2], [13] and [14]. As a particular case, we analyze the use of scalar and matrix-valued parameters in a Lp(Ω)M context. We apply the results developed to some boundary value problems involving the p-Laplacian operator, where it is shown that the use of matrix-valued parameters improves the convergence of the algorithm.

An explicit convergent finite difference scheme for motion of level sets by mean curvature is presented. The scheme is defined on a cartesian grid, using neighbors arranged approximately in a circle. The accuracy of the scheme, which depends on the radius of the circle, dx, and on the angular resolution, dθ, is formally O(dx2+dθ). The scheme is explicit and nonlinear: the update involves computing the median of the values at the neighboring grid points. Numerical results suggest that despite the low accuracy, acceptable results are achieved for small stencil sizes. A numerical example is presented which shows that the centered difference scheme is non-convergent.

Solving a variational inequality problem VI(Ω, F) is equivalent to finding a solution of a system of nonsmooth equations (a hard problem). The Peaceman-Rachford and /or Douglas-Rachford operator splitting methods are advantageous when they are applied to solve variational inequality problems, because they solve the original problem via solving a series of systems of nonlinear smooth equations (a series of easy problems). Although the solution of VI(Ω, F) is invariant under multiplying F by some positive scalar β, yet the numerical experiment has shown that the number of iterations depends significantly on the positive parameter β which is a constant in the original operator splitting methods. In general, it is difficult to choose a proper parameter β for individual problems. In this paper, we present a modified operator splitting method which adjusts the scalar parameter automatically per iteration based on the message of the iterates. Exact and inexact forms of the modified method with self-adaptive variable parameter are suggested and proved to be convergent under mild assumptions. Finally, preliminary numerical tests show that the self-adaptive adjustment rule is proper and necessary in practice.

The Schwarz Alternating Method can be used to solve elliptic boundary value problems on domains which consist of two or more overlapping subdomains. The solution is approximated by an infinite sequence of functions which results from solving a sequence of elliptic boundary value problems in each subdomain. In this paper, proofs of convergence of some Schwarz Alternating Methods for nonlinear elliptic problems which are known to have solutions by the monotone method (also known as the method of subsolutions and supersolutions) are given. In particular, an additive Schwarz method for scalar as well some coupled nonlinear PDEs are shown to converge to some solution on finitely many subdomains, even when multiple solutions are possible. In the coupled system case, each subdomain PDE is linear, decoupled and can be solved concurrently with other subdomain PDEs. These results are applicable to several models in population biology.

In this paper we establish two new projection-type methods for the solution of monotone linear complementarity problem (LCP). The methods are a combination of the extragradient method and the Newton method, in which the active set strategy is used and only one linear system of equations with lower dimension is solved at each iteration. It is shown that under the assumption of monotonicity, these two methods are globally and linearly convergent. Furthermore, under a nondegeneracy condition they have a finite termination property. At last, the methods are extended to solving monotone affine variational inequality problem.

We consider the fast solution of a class of large, piecewise smooth minimization problems. For lack of smoothness, usual Newton multigrid methods cannot be applied. We propose a new approach based on a combination of convex minization with constrained Newton linearization. No regularization is involved. We show global convergence of the resulting monotone multigrid methods and give polylogarithmic upper bounds for the asymptotic convergence rates. Efficiency is illustrated by numerical experiments.

We analyze in the $L^\infty-$ norm a class of semi-Lagrangian advective schemes introduced by the author and A. Staniforth in 1992 to improve the solution produced by numerical models for weather prediction and climate studies that use semi-Lagrangian advective schemes. The new quasi-monotone and conservative semi-Lagrangian schemes are $L^\infty-$ stable and converge optimally when the solution is sufficiently smooth.

In this paper, we discuss semiconvergence of the matrix splitting methods for solving singular linear systems. The concepts that a splitting of a matrix is regular or nonnegative are generalized and we introduce the terminologies that a splitting is quasi-regular or quasi-nonnegative. The equivalent conditions for the semiconvergence are proved. Comparison theorem on convergence factors for two different quasi-nonnegative splittings is presented. As an application, the semiconvergence of the power method for solving the Markov chain is derived. The monotone convergence of the quasi-nonnegative splittings is proved. That is, for some initial guess, the iterative sequence generated by the iterative method introduced by a quasi-nonnegative splitting converges towards a solution of the system from below or from above.

Infeasible-interior-point paths \(\{(x(r,\eta),y(r,\eta))\}_{r>0}\), \(0 0, y>0, x_iy_i &=& r\eta_i, & 1\le i\le n. \end{array} \leqno(LCP)_r \) If the path \(z(r,\eta)=(x(r,\eta),y(r,\eta))\) converges for \(r\downarrow 0\), then it converges to a solution of \((LCP)_0\). This paper deals with the analyticity properties of \(z(r,\eta)\) and its derivatives with respect to r near r = 0 for solvable monotone complementarity problems \((LCP)_0\). It is shown for \((LCP)_0\) with a strictly complementary solution that the path \(z(r,\eta)\), \(r\downarrow0\), has an extension to \(r=0\) which is analytic also at \(r=0\). If \((LCP)_0\) has no strictly complementary solution, then \(\hat z(\rho,\eta):=z(\rho^2,\eta)\), \(\rho=\sqrt r\), has an extension to \(\rho=0\) that is analytic at \(\rho=0\).

This paper is concerned with numerical methods for a finite difference system of reaction-diffusion-convection equation under nonlinear boundary condition. Various monotone iterative methods are presented, and each of these methods leads to an existence-comparison theorem as well as a computational algorithm for numerical solutions. The monotone property of the iterations gives improved upper and lower bounds of the solution in each iteration, and the rate of convergence of the iterations is either quadratic or nearly quadratic depending on the property of the nonlinear function. Application is given to a model problem from chemical engineering, and some numerical results, including a test problem with known analytical solution, are presented to illustrate the various rates of convergence of the iterations.

Given a nonsingular matrix $A$ , and a matrix $T$ of the same order, under certain very mild conditions, there is a unique splitting $A=B-C$ , such that $T=B^{-1}C$ . Moreover, all properties of the splitting are derived directly from the iteration matrix $T$ . These results do not hold when the matrix $A$ is singular. In this case, given a matrix $T$ and a splitting $A=B-C$ such that $T=B^{-1}C$ , there are infinitely many other splittings corresponding to the same matrices $A$ and $T$ , and different splittings can have different properties. For instance, when $T$ is nonnegative, some of these splittings can be regular splittings, while others can be only weak splittings. Analogous results hold in the symmetric positive semidefinite case. Given a singular matrix $A$ , not for all iteration matrices $T$ there is a splitting corresponding to them. Necessary and sufficient conditions for the existence of such splittings are examined. As an illustration of the theory developed, the convergence of certain alternating iterations is analyzed. Different cases where the matrix is monotone, singular, and positive (semi)definite are studied.

Solutions of symmetric Riccati differential equations (RDEs for short) are in the usual applications positive semidefinite matrices. Moreover, in the class of semidefinite matrices, solutions of different RDEs are also monotone, with respect to properly ordered data. Positivity and monotonicity are essential properties of RDEs. In Dieci and Eirola (1994), we showed that, generally, a direct discretization of the RDE cannot maintain positivity, and be of order greater than one. To get higher order, and to maintain positivity, we are thus forced to look into indirect solution procedures. Here, we consider the problem of how to maintain monotonicity in the numerical solutions of RDEs. Naturally, to obtain order greater than one, we are again forced to look into indirect solution procedures. Still, the restrictions imposed by monotonicity are more stringent that those of positivity, and not all of the successful indirect solution procedures of Dieci and Eirola (1994) maintain monotonicity. We prove that by using symplectic Runge-Kutta (RK) schemes with positive weights (e.g., Gauss schemes) on the underlying Hamiltonian matrix, we eventually maintain monotonicity in the computed solutions of RDEs.

We derive globally convergent multigrid methods for discrete elliptic variational inequalities of the second kind as obtained from the approximation of related continuous problems by piecewise linear finite elements. The coarse grid corrections are computed from certain obstacle problems. The actual constraints are fixed by the preceding nonlinear fine grid smoothing. This new approach allows the implementation as a classical V-cycle and preserves the usual multigrid efficiency. We give $1-O(j^{-3})$ estimates for the asymptotic convergence rates. The numerical results indicate a significant improvement as compared with previous multigrid approaches.

Two block monotone iterative schemes for a nonlinear algebraic system, which is a finite difference approximation of a nonlinear elliptic boundary-value problem, are presented and are shown to converge monotonically either from above or from below to a solution of the system. This monotone convergence result yields a computational algorithm for numerical solutions as well as an existence-comparison theorem of the system, including a sufficient condition for the uniqueness of the solution. An advantage of the block iterative schemes is that the Thomas algorithm can be used to compute numerical solutions of the sequence of iterations in the same fashion as for one-dimensional problems. The block iterative schemes are compared with the point monotone iterative schemes of Picard, Jacobi and Gauss-Seidel, and various theoretical comparison results among these monotone iterative schemes are given. These comparison results demonstrate that the sequence of iterations from the block iterative schemes converges faster than the corresponding sequence given by the point iterative schemes. Application of the iterative schemes is given to a logistic model problem in ecology and numerical ressults for a test problem with known analytical solution are given.

We obtain estimates on the minimal size of an equally spaced grid which can be used to construct monotone and consistent approximations to elliptic problems. The estimates are given in terms of the ellipticity of the equation. This problem is related to diophantine approximations, see the Appendix written by W. M. Schmidt.

The finite element method for an elliptic equation with discontinuous coefficients (obtained for the magnetic potential from Maxwell's equations) is analyzed in the union of closed domains the boundaries of which form a system of three circles with the same centre. As the middle domain is very narrow the triangulations obeying the maximum angle condition are considered. In the case of piecewise linear trial functions the maximum rate of convergence $O(h)$ in the norm of the space $H^1(\Omega_h)$ is proved under the following conditions: 1. the exact solution $u\in H^1(\Omega)$ is piecewise of class $H^2$ ; 2. the family of subtriangulations $\{{\cal T}_h^{\rm A}\}$ of the narrow subdomain $\Omega^{\rm A}$ satisfies the maximum angle condition expressed by relation (38). The paper extends the results of [24].

This paper presents general algorithms for the parallel solution of finite element problems associated with maximal monotone operators of local type. The latter concept, which is also introduced here, is well suited to capture the idea that the given operator is the discretization of a differential operator that may involve nonlinearities and/or constraints as long as those are of a local nature. Our algorithms are obtained as a combination of known algorithms for possibly multi-valued maximal monotone operators with appropriate decompositions of the domain. This work extends a method due to two of the authors in the single-valued and linear case.

We derive fast solvers for discrete elliptic variational inequalities of the first kind (obstacle problems) as resulting from the approximation of related continuous problems by piecewise linear finite elements. Using basic ideas of successive subspace correction, we modify well-known relaxation methods by extending the set of search directions. Extended underrelaxations are called monotone multigrid methods, if they are quasioptimal in a certain sense. By construction, all monotone multigrid methods are globally convergent. We take a closer look at two natural variants, the standard monotone multigrid method and a truncated version. For the considered model problems, the asymptotic convergence rates resulting from the standard approach suffer from insufficient coarse-grid transport, while the truncated monotone multigrid method provides the same efficiency as in the unconstrained case.

Parallel block two-stage iterative methods for the solution of linear systems of algebraic equations are studied. Convergence is shown for monotone matrices and for H-matrices. Two different asynchronous versions of these methods are considered and their convergence investigated.

We present a LagrangeC 2-interpolant to scattered convex data which preserves convexity. We also present a LagrangeC 2-interpolant to uniformly spaced monotone data sites which preserves monotonicity. In both cases no further conditions are required on the data values. These interpolants are explicitely described and local. Error isO(h 3) when the function to be interpolated isC 3.

Convergence of two-stage iterative methods for the solution of linear systems is studied. Convergence of the non-stationary method is shown if the number of inner iterations becomes sufficiently large. TheR 1-factor of the two-stage method is related to the spectral radius of the iteration matrix of the outer splitting. Convergence is further studied for splittings ofH-matrices. These matrices are not necessarily monotone. Conditions on the splittings are given so that the two-stage method is convergent for any number of inner iterations.

A functionf ?C (Ω), $$\Omega \subseteq \mathbb{R}^s $$ is called monotone on Ω if for anyx, y ? Ω the relation x ? y ? ? + s impliesf(x)?f(y). Given a domain $$\Omega \subseteq \mathbb{R}^s $$ with a continuous boundary ?Ω and given any monotone functionf on ?Ω we are concerned with the existence and regularity ofmonotone extensions i.e., of functionsF which are monotone on all of Ω and agree withf on ?Ω. In particular, we show that there is no linear mapping that is capable of producing a monotone extension to arbitrarily given monotone boundary data. Three nonlinear methods for constructing monotone extensions are then presented. Two of these constructions, however, have the common drawback that regardless of how smooth the boundary data may be, the resulting extensions will, in general, only be Lipschitz continuous. This leads us to consider a third and more involved monotonicity preserving extension scheme to prove that, when Ω is the unit square [0, 1]2 in ?2, strictly monotone analytic boundary data admit a monotone analytic extension.

Theorem.Let the sequences {e i (n) },i=1, 2, 3,n=0, 1, 2, ...be defined by[Figure not available: see fulltext.]where the e (0)? s satisfy[Figure not available: see fulltext.]and where all square roots are taken positive. Then[Figure not available: see fulltext.]where the convergence is quadratic and monotone and where[Figure not available: see fulltext.] The discussions of convergence are entirely elementary. However, although the determination of the limits can be made in an elementary way, an acquaintance with elliptic objects is desirable for real understanding. In fact thee i (0) 's will be interpreted as the values of a ?-function at its half periods and the successive e i (n)? s will be the corresponding values for a ?-function with the same real period and with the imaginary period doubled at each stage-this is the Landen transformation. Ultimately the ?-function will degenerate into a trigonometrical function. The subtitle is explained by the fact that the real half-period, ?, of the ?-function defined by $$y^{'2} = 4(y - e_1^{(0)} )(y - e_2^{(0)} )(y - e_3^{(0)} ) if found from W = \frac{1}{{12}}\left( {\frac{\pi }{\omega }} \right)^2 $$ .

Comparison theorems for weak splittings of bounded operators are presented. These theorems extend the classical comparison theorem for regular splittings of matrices by Varga, the less known result by Wo?nicki, and the recent results for regular and weak regular splittings of matrices by Neumann and Plemmons, Elsner, and Lanzkron, Rose and Szyld. The hypotheses of the theorems presented here are weaker and the theorems hold for general Banach spaces and rather general cones. Hypotheses are given which provide strict inequalities for the comparisons. It is also shown that the comparison theorem by Alefeld and Volkmann applies exclusively to monotone sequences of iterates and is not equivalent to the comparison of the spectral radius of the iteration operators.

In this paper we introduce the set of so-called monotone iteration functions (MI-functions) belonging to a given function. We prove necessary and sufficient conditions in order that a given MI-function is (in a precisely defined sense) at least as fast as a second one. Regular splittings of a function which were initially introduced for linear functions by R.S. Varga in 1960 are generating MI-functions in a natural manner. For linear functions every MI-function is generated by a regular splitting. For nonlinear functions, however, this is generally not the case.

We consider the approximation of the solutions of the integral equations of the form $$\lambda x(s) = \int\limits_a^b {K(s,t,x(t))dt + y(s)} ,a \leqq s \leqq b,$$ or of the form $$x(s) = \int\limits_a^b {K(s,t)f(t,x(t))dt + y(s)} ,a \leqq s \leqq b$$ by using the techniques of monotone operators.

Subject to rather mild assumptions on the integrand, the Romberg table (theT-table) and the modified Romberg table (theU-table) yield asymptotically upper and lower bounds for the value of the integral, and the convergence of the columns of the two tables is asymptotically monotone. This is verified for arbitrary sequences of step sizes satisfying the usual condition of convergence for Romberg integration.

The stability, accuracy, and convergence of the basic fractional step algorithms are analyzed when these algorithms are used to compute discontinuous solutions of scalar conservation laws. In particular, it is proved that both first order splitting and Strang splitting algorithms always converge to the unique weak solution satisfying the entropy condition. Examples of discontinuous solutions are presented where both Strang-type splitting algorithms are only first order accurate but one of the standard first order algorithms is infinite order accurate. Various aspects of the accuracy, convergence, and correct entropy production are also studied when each split step is discretized via monotone schemes, Lax-Wendroff schemes, and the Glimm scheme.

This paper presents an existence-comparison theorem and an iterative method for a nonlinear finite difference system which corresponds to a class of semilinear parabolic and elliptic boundary-value problems. The basic idea of the iterative method for the computation of numerical solutions is the monotone approach which involves the notion of upper and lower solutions and the construction of monotone sequences from a suitable linear discrete system. Using upper and lower solutions as two distinct initial iterations, two monotone sequences from a suitable linear system are constructed. It is shown that these two sequences converge monotonically from above and below, respectively, to a unique solution of the nonlinear discrete equations. This formulation leads to a well-posed problem for the nonlinear discrete system. Applications are given to several models arising from physical, chemical and biological systems. Numerical results are given to some of these models including a discussion on the rate of convergence of the monotone sequences.

In this paper we reanalyze the trapezoidal method for the solution of nonlinear Abel-Volterra integral equations on the half line. We prove the convergence of the method in the uniform norm, provided the nonlinearity is Lipschitz-continuous and strictly monotone.

In this paper, we consider monotone explicit iterations of the finite element schemes for the nonlinear equations associated with the boundary value problem Δu=bu 2, based on piecewise linear polynomials and the lumping operator. These iterations construct the monotonically decreasing and increasing sequences, and convergence proofs are given. Finally, we present some numerical examples verifying the effectiveness of the theory.

Some hyperbolic differential operators of two dimensions are proved to be of monotone kind (see [2], p. 276), and it is shown how to use this property to enclose solutions of hyperbolic initial value problems by iteration methods with monotonicity properties. The application of the theorems is demonstrated by numerical examples.

A parabolic problem of the following form is considered (1) $$\frac{\partial }{{\partial t}}\left[ {a\vartheta + w} \right] - \Delta \vartheta = f$$ (2) $$w \varepsilon \Lambda (\vartheta ),$$ wherea is a positive constant,f is a datum and λ is a maximal monotone graph. This system contains the (weak formulation of the)Stefan problem as a particular case. Here the problem (1), (2) is approximated by coupling (1) with therelaxed equation (3) $$\varepsilon \frac{{\partial w}}{{\partial t}} + \Lambda ^{ - 1} (w) \mathrel\backepsilon \vartheta (\varepsilon : constant > 0).$$ The problem (1), (3) is then discretized in time by thesemi-explicit scheme (4) $$a\frac{{\vartheta ^n - \vartheta ^{n - 1} }}{\tau } + \frac{{w^n - w^{n - 1} }}{\tau } - \Delta \vartheta ^n = f^n$$ (5) $$\varepsilon \frac{{w^n - w^{n - 1} }}{\tau } + \Lambda ^{ - 1} (w^n ) \mathrel\backepsilon \vartheta ^{n - 1} ;$$ a finite element space discretization and quadrature formulae are then introduced. Thus at each time-step (5) is replaced by a finite number ofindependent algebraic equations, which can be solved with respect to the barycentral values ofw n ; then (4) is reduced to alinear system of algebraic equations having as unknowns the nodal values of ϑ n . Assuming the condition τ/e≦a, the fully discrete scheme is stable and its solution converges to that of (1), (2). Error estimates are proved. The results of some numerical experiments are discussed; they show that the present method is faster than other classical procedures.

In a previous paper [6] we considered the problem of approximating (in the Chebyshev-norm) a real-valued functionf(x) on a compact subsetX m ,m?1, by an element of a set of functionsa(p, x),p?P,P

LetQ n be the quadrature rule of Gauss or Newton-Cotes withn abscissas. It is proven here, thatf (2n)?0 impliesQ n G [f]?Q m G [f] (for allm>n) andQ 2n?1 NC [f]?Q 2n NC [f]?Q 2n+1 NC [f]. It follows that the sequenceQ n[f] (n=1, 2, ...) is monotone, if all derivatives off are positive.

Let $$f(z) = \sum\limits_{j = 0}^\infty{t_{2j} z^{2j} } ,t_{2j}\geqq 0(j = 0,1,2,...)$$ , be holomorphic in an open disc with the centrez0=0 and radiusr>1. LetQn (n=1, 2, ...) be interpolatory quadrature formulas approximating the integral $$\int\limits_{ - 1}^{ + 1} {f(x)dx} $$ . In this paper some classes of interpolatory quadratures are considered, which are based on the zeros of orthogonal polynomials corresponding to an even weight function. It is shown that the sequencesQn9f] (n=1, 2, ...) are monotone. Especially we will prove monotony in Filippi's quadrature rule and with an additional assumption onf monotony in the Clenshaw-Curtis quadrature rule.

A difference scheme of exponential type for solving a nonlinear singular perturbation problem is analysed. Although this scheme is not of monotone type, aL 1 convergence result is obtained. Relations between this scheme and Engquist-Osher scheme are also discussed.

In this paper we establish the stability of the Ritz procedure for the nonlinear equationAu+F(u)?f=0 whereA is a linear positive definite operator in a real Hilbert spaceH andF is a monotone nonlinear operator.

We study monotone iterative methods for the numerical solution of a class of nonlinear elliptic boundary value problems by applying a theorem of Ortega and Rheinboldt. This generalizes the work of earlier authors to the case of nonlinear perturbations of linear problems with a nontrivial kernel.

Consider the following quasilinear elliptic PDE, which is equivalent to a nonlinear variational inequality: ?divF(?u)+β(u)?f. Here β is a singular maximal monotone graph and the nonlinear differential operator is only assumed to be monotone; surfaces of prescribed mean curvature over obstacles may thus be viewed as relevant examples. The numerical approximation proposed in this paper consists of combining continuous piecewise linear finite elements with a preliminary regularization of β. The resulting scheme is shown to be quasi-optimally accurate inL ?. The underlying analysis makes use of both a topological technique and a sharpL p -duality argument.