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     

Second-order schemes for conservation laws with discontinuous flux modelling clarifier–thickener units

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 (Bürger, 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.     

A note on the first‐order logic of complete BL‐chains

In [10] it is claimed that the set of predicate tautologies of all complete BL‐chains and the set of all standard tautologies (i. e., the set of predicate formulas valid in all standard BL‐algebras) coincide. As noticed in [11], this claim is wrong. In this paper we show that a complete BL‐chain B satisfies all standard BL‐tautologies iff for any transfinite sequence (a_i: i ∈ I) of elements of B , the condition ∧_{i ∈ I} (a²_i ) = (∧_{i ∈ I} a_i)² holds in B . (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Existence in the Large for Pressure-Gradient System

In this paper, the authors use Glimm scheme to study the global existence of BV solutions to Cauchy problem of the pressure-gradient system with large initial data. To this end, some important properties of the shock curves of the pressure-gradient system in the Riemann invariant coordinate system and verify that the shock curves satisfy Diperna’s conditions (see [Diperna, R. J., Existence in the large for quasilinear hyperbolic conservation laws, Arch. Ration. Mech. Anal., 52(3), 1973, 244–257]) are studied. Then they construct the approximate solution sequence through Glimm scheme. By establishing accurate local interaction estimates, they prove the boundedness of the approximate solution sequence and its total variation.     

Local discontinuous Galerkin methods with explicit Runge‐Kutta time marching for nonlinear carburizing model

A fully discrete local discontinuous Galerkin (LDG) method coupled with 3 total variation diminishing Runge‐Kutta time‐marching schemes, for solving a nonlinear carburizing model, will be analyzed and implemented in this paper. On the basis of a suitable numerical flux setting in the LDG method, we obtain the optimal error estimate for the Runge‐Kutta–LDG schemes by energy analysis, under the condition τ ≤ λh², where h and τ are mesh size and time step, respectively, λ is a positive constant independent of h. Numerical experiments are presented to verify the accuracy and capability of the proposed schemes. For the carburizing diffusion processes of steel and the diffusion simulation for Cu‐Ni system, the numerical results show good agreement with the experimental results.     

Superconvergence of the direct discontinuous Galerkin method for a time‐fractional initial‐boundary value problem

A time‐fractional reaction–diffusion initial‐boundary value problem with periodic boundary condition is considered on Q ? Ω × [0, T] , where Ω is the interval [0, l] . Typical solutions of such problem have a weak singularity at the initial time t = 0. The numerical method of the paper uses a direct discontinuous Galerkin (DDG) finite element method in space on a uniform mesh, with piecewise polynomials of degree k ≥ 2 . In the temporal direction we use the L1 approximation of the Caputo derivative on a suitably graded mesh. We prove that at each time level of the mesh, our L1‐DDG solution is superconvergent of order k + 2 in L²(Ω) to a particular projection of the exact solution. Moreover, the L1‐DDG solution achieves superconvergence of order (k + 2) in a discrete L²(Q) norm computed at the Lobatto points, and order (k + 1) superconvergence in a discrete H¹(Q) seminorm at the Gauss points; numerical results show that these estimates are sharp.     

A characterization of periodic solutions for time‐fractional differential equations in UMD spaces and applications

We study the fractional differential equation (*) D^αu(t) + BD^βu(t) + Au(t) = f(t), 0 ? t ? 2π (0 ? β < α ? 2) in periodic Lebesgue spaces L^p(0, 2π; X) where X is a Banach space. Using functional calculus and operator valued Fourier multiplier theorems, we characterize, in UMD spaces, the well posedness of (*) in terms of R‐boundedness of the sets {(ik)^α((ik)^α + (ik)^βB + A)^?1}_{k∈ Z} and {(ik)^βB((ik)^α + (ik)^βB + A)^?1}_{k∈ Z} . Applications to the fractional problems with periodic boundary condition, which includes the time diffusion and fractional wave equations, as well as an abstract version of the Basset‐Boussinesq‐Oseen equation are treated. © 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim     

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     

The unconditional stability and mass‐preserving S‐DDM scheme for solving parabolic equations in three dimensions

In this paper, the unconditional stability and mass‐preserving splitting domain decomposition method (S‐DDM) for solving three‐dimensional parabolic equations is analyzed. At each time step level, three steps (x‐direction, y‐direction, and z‐direction) are proposed to compute the solutions on each sub‐domains. The interface fluxes are first predicted by the semi‐implicit flux schemes. Second, the interior solutions and fluxes are computed by the splitting implicit solution and flux coupled schemes. Last, we recompute the interface fluxes by the explicit schemes. Due to the introduced z‐directional splitting and domain decomposition, the analysis of stability and convergence is scarcely evident and quite difficult. By some mathematical technique and auxiliary lemmas, we prove strictly our scheme meet unconditional stability and give the error estimates in L²‐norm. Numerical experiments are presented to illustrate the theoretical analysis.     

Commutators of generalized Hardy operators

LetH_a,b be the commutator generated by the generalized Hardy operator and the CMO function. The (L^p, L^p) boundedness of H_a,b is discussed in this paper. Furthermore, the authors consider the boundedness of H_a,b on the weighted homogeneous Herz spaces (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Well-posedness in <Emphasis Type="Italic">BV</Emphasis><Subscript><Emphasis Type="Italic">t</Emphasis></Subscript> and convergence of a difference scheme for continuous sedimentation in ideal clarifier-thickener units

Summary. We consider a scalar conservation law modeling the settling of particles in an ideal clarifier-thickener unit. The conservation law has a nonconvex flux which is spatially dependent on two discontinuous parameters. We suggest to use a Krukov-type notion of entropy solution for this conservation law and prove uniqueness (L¹ stability) of the entropy solution in the BV_t class (functions W(x,t) with _tW being a finite measure). The existence of a BV_t entropy solution is established by proving convergence of a simple upwind finite difference scheme (of the Engquist-Osher type). A few numerical examples are also presented.Mathematics Subject Classification (2000):35L65, 35R05, 65M06, 76T20     

On the Convergence of Finite Element Method for Second Order Elliptic Interface Problems

The purpose of this paper is to study the convergence of finite element approximation to the exact solution of general self-adjoint elliptic equations with discontinuous coefficients. Due to low global regularity of the solution, it is difficult to achieve optimal order of convergence with classical finite element methods [Numer. Math. 1998; 79:175–202]. In this paper, an isoparametric type of discretization is used to prove optimal order error estimates in L ² and H ¹ norms when the global regularity of the solution is low. The interface is assumed to be of arbitrary shape and is smooth for our purpose. Further, for the purpose of numerical computations, we discuss the effect of numerical quadrature on finite element solution, and the related optimal order estimates are also established.     

Regularity of minimizers of semilinear elliptic problems up to dimension 4

We consider the class of semistable solutions to semilinear equations ?Δu = f(u) in a bounded smooth domain Ω of \input amssym $\Bbb R^n$ (with Ω convex in some results). This class includes all local minimizers, minimal, and extremal solutions. In dimensions n ≤ 4, we establish an a priori L^∞‐bound that holds for every positive semistable solution and every nonlinearity f. This estimate leads to the boundedness of all extremal solutions when n = 4 and Ω is convex. This result was previously known only in dimensions n ≤ 3 by a result of G. Nedev. In dimensions 5 ≤ n ≤ 9 the boundedness of all extremal solutions remains an open question. It is only known to hold in the radial case Ω = B_R by a result of A. Capella and the author. © 2010 Wiley Periodicals, Inc.     

Lp (ℝn ) boundedness for higher commutators of singular integrals with rough kernels belonging to certain block spaces

In this paper, we prove the L^p (?ⁿ ) boundedness for higher commutators of singular integrals with rough kernels belonging to certain block spaces provided that 1 < p < ∞ (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

l 1-error estimates on the immersed interface upwind scheme for linear convection equations with piecewise constant coefficients: A simple proof

Abstract A linear convection equation with discontinuous coefcients arises in wave propagation through interfaces.An interface condition is needed at the interface to select a unique solution.An upwind scheme that builds this interface condition into its numerical flux is called the immersed interface upwind scheme.An l1-error estimate of such a scheme was frst established by Wen et al.(2008).In this paper,we provide a simple analysis on the l1-error estimate.The main idea is to formulate the solution to the underline initial-value problem into the sum of solutions to two convection equations with constant coefcients,which can then be estimated using classical methods for the initial or boundary value problems.     

Boundary behavior of Gleason''''s problem in hyperbolic harmonic Bergman spaces

Necessary and sufficient conditions are obtained for the boundedness of Berezin transformation on Lebesgue space Lp(B, dVβ) in the real unit ball B in Rn. As an application, we prove that Gleason type problem is solvable in hyperbolic harmonic Bergman spaces. Furthermore we investigate the boundary behavior of the solutions of Gleason type problem.     

On the decay and stability of global solutions to the 3‐D inhomogeneous Navier‐Stokes equations

In this paper, we investigate the large‐time decay and stability to any given global smooth solutions of the 3‐D incompressible inhomogeneous Navier‐Stokes equations. In particular, we prove that given any global smooth solution (a,u) of (1.2), the velocity field u decays to 0 with an explicit rate, which coincides with the L² norm decay for the weak solutions of the 3‐D classical Navier‐Stokes system [26,29] as t goes to ∞. Moreover, a small perturbation to the initial data of (a,u) still generates a unique global smooth solution to (1.2), and this solution keeps close to the reference solution (a,u) for t > 0. We should point out that the main results in this paper work for large solutions of (1.2). © 2010 Wiley Periodicals, Inc.     

Well‐posedness for the incompressible magneto‐hydrodynamic system

This paper is concerned with well‐posedness of the incompressible magneto‐hydrodynamics (MHD) system. In particular, we prove the existence of a global mild solution in BMO^?1 for small data which is also unique in the space C([0, ∞); BMO^?1). We also establish the existence of a local mild solution in bmo^?1 for small data and its uniqueness in C([0, T); bmo^?1). In establishing our results an important role is played by the continuity of the bilinear form which was proved previously by Kock and Tataru. In this paper, we give a new proof of this result by using the weighted L^p‐boundedness of the maximal function. Copyright © 2006 John Wiley & Sons, Ltd.     

Least‐squares solutions and least‐rank solutions of the matrix equation AXA* = B and their relations

A Hermitian matrix X is called a least‐squares solution of the inconsistent matrix equation AXA^* = B, where B is Hermitian. A^* denotes the conjugate transpose of A if it minimizes the F‐norm of B ? AXA^*; it is called a least‐rank solution of AXA^* = B if it minimizes the rank of B ? AXA^*. In this paper, we study these two types of solutions by using generalized inverses of matrices and some matrix decompositions. In particular, we derive necessary and sufficient conditions for the two types of solutions to coincide. Copyright © 2012 John Wiley & Sons, Ltd.     

Nonlinear perturbations of linear accretive operators in Banach spaces

It is shown that a strongly continuous semi-group of nonlinear nonexpansive operators can be constructed as lim _n→∞ ((I+t/nB)⁻¹ (I+t/nB)⁻¹) ⁿ whereA is a linearm-accretive operator,B is a nonlinearm-accretive operator, andB satisfies a boundedness condition relative toA.