A Note on Gradient Estimates for Elliptic Equations with Discontinuous Coefficients

The authors will use a method in metric geometry to show an L^(P)-estimate for gradient of the weak solutions to elliptic equations with discontinuous coefficients,even the BMO semi-norms of the coefficients are not small.They also extend them to the weak solutions to parabolic equations.

Uniform Convergence of Multigrid V-Cycle on Adaptively Refined Finite Element Meshes for Elliptic Problems with Discontinuous Coefficients

The multigrid V-cycle methods for adaptive finite element discretizations of two-dimensional elliptic problems with discontinuous coefficients are considered.Under the conditions that the coefficient is quasi-monotone up to a constant and the meshes are locally refined by using the newest vertex bisection algorithm,some uniform convergence results are proved for the standard multigrid V-cycle algorithm with Gauss-Seidel relaxations performed only on new nodes and their immediate neighbours.The multigrid V-cycle algorithm uses O(N)operations per iteration and is optimal.

Second Order Multigrid Methods for Elliptic Problems with Discontinuous Coefficients on an Arbitrary Interface, I: One Dimensional Problems

In this paper we present a one dimensional second order accurate method to solve Elliptic equations with discontinuous coefficients on an arbitrary interface.Second order accuracy for the first derivative is obtained as well.The method is based on the Ghost Fluid Method,making use of ghost points on which the value is defined by suitable interface conditions.The multi-domain formulation is adopted,where the problem is split in two sub-problems and interface conditions will be enforced to close the problem.Interface conditions are relaxed together with the internal equations(following the approach proposed in[10]in the case of smooth coefficients),leading to an iterative method on all the set of grid values(inside points and ghost points).A multigrid approach with a suitable definition of the restriction operator is provided.The restriction of the defect is performed separately for both sub-problems,providing a convergence factor close to the one measured in the case of smooth coefficient and independent on the magnitude of the jump in the coefficient.Numerical tests will confirm the second order accuracy.Although the method is proposed in one dimension,the extension in higher dimension is currently underway[12]and it will be carried out by combining the discretization of[10]with the multigrid approach of[11]for Elliptic problems with non-eliminated boundary conditions in arbitrary domain.

ASYMPTOTIC BEHAVIOUR OF EIGENVALUES FOR THE DISCONTINUOUS BOUNDARY-VALUE PROBLEM WITH FUNCTIONAL-TRANSMISSION CONDITIONS

In this study, the boundary-value problem with eigenvalue parameter generated by the differential equation with discontinuous coefficients and boundary conditions which contains not only endpoints of the considered interval, but also point of discontinuity and linear functionals is investigated. So, the problem is not pure boundary-value. The authors single out a class of linear functionals and find simple algebraic conditions on coefficients, which garantee the existence of infinit number eigenvalues. Also the asymptotic formulas for eigenvalues are found.

LOCAL MULTILEVEL METHODS FOR SECOND-ORDER ELLIPTIC PROBLEMS WITH HIGHLY DISCONTINUOUS COEFFICIENTS

In this paper, local multiplicative and additive multilevel methods on adaptively refined meshes are considered for second-order elliptic problems with highly discontinuous coeffi- cients. For the multilevel-preconditioned system, we study the distribution of its spectrum by using the abstract Schwarz theory. It is proved that, except for a few small eigenval- ues, the spectrum of the preconditioned system is bounded quasi-uniformly with respect to the jumps of the coefficient and the mesh sizes. The convergence rate of multilevel- preconditioned conjugate gradient methods is shown to be quasi-optimal regarding the jumps and the meshes. Numerical experiments are presented to illustrate the theoretical findings.

Extrapolation Cascadic Multigrid Method for Cell-Centered FV Discretization of Diffusion Equations with Strongly Discontinuous and Anisotropic Coefficients

Extrapolation cascadic multigrid(EXCMG)method with conjugate gradient smoother is very efficient for solving the elliptic boundary value problems with linearfinite element discretization.However,it is not trivial to generalize the vertex-centred EXCMG method to cell-centeredfinite volume(FV)methods for diffusion equations with strongly discontinuous and anisotropic coefficients,since a non-nested hierarchy of grid nodes are used in the cell-centered discretization.For cell-centered FV schemes,the vertex values(auxiliary unknowns)need to be approximated by cell-centered ones(primary unknowns).One of the novelties is to propose a new gradient transfer(GT)method of interpolating vertex unknowns with cell-centered ones,which is easy to implement and applicable to general diffusion tensors.The main novelty of this paper is to design a multigrid prolongation operator based on the GT method and splitting extrapolation method,and then propose a cell-centered EXCMG method with BiCGStab smoother for solving the large linear system resulting from linear FV discretization of diffusion equations with strongly discontinuous and anisotropic coefficients.Numerical experiments are presented to demonstrate the high efficiency of the proposed method.

THE FINITE ELEMENT METHOD FOR SEMILINEARPARABOLIC EQUATIONS WITH DISCONTINUOUSCOEFFICIENTS

In this paper we investigate the existence, uniqueness and regularity of the solution of semilinear parabolic equations with coefficients that are discontinuous across the interface, some prior estimates are obtained. A net shape of the finite elements around the singular points was designed in [7] to solve the linear elliptic problems, by means of that net, we prove that the approximate solution has the same convergence rate as that without singularity.

Mean-field stochastic differential equations with a discontinuous diffusion coefficient

We study R^(d)-valued mean-field stochastic differential equations with a diffusion coefficient that varies in a discontinuous manner on the L_(p)-norm of the process.We establish the existence of a unique global strong solution in the presence of a robust drift,while also investigating scenarios where the presence of a global solution is not assured.

A New Discontinuous Galerkin Method for Parabolic Equations with Discontinuous Coefficient

In this paper,a new discontinuous Galerkin method is developed for the parabolic equation with jump coefficients satisfying the continuous flow condition.Theoretical analysis shows that this method is L^(2) stable.When the finite element space consists of interpolative polynomials of degrees k,the convergent rate of the semi-discrete discontinuous Galerkin scheme has an order of δ(h^(k)).Numerical examples for both 1-dimensional and 2-dimensional problems demonstrate the validity of the new method.

A Uniformly Stable Nonconforming FEM Based on Weighted Interior Penalties for Darcy-Stokes-Brinkman Equations

A nonconforming rectangular finite element method is proposed to solve a fluid structure interaction problem characterized by the Darcy-Stokes-Brinkman Equation with discontinuous coefficients across the interface of different structures.A uniformly stable mixed finite element together with Nitsche-type matching condi-tions that automatically adapt to the coupling of different sub-problem combinations are utilized in the discrete algorithm.Compared with other finite element methods in the literature,the new method has some distinguished advantages and features.The Boland-Nicolaides trick is used in proving the inf-sup condition for the multi-domain discrete problem.Optimal error estimates are derived for the coupled prob-lem by analyzing the approximation errors and the consistency errors.Numerical examples are also provided to confirm the theoretical results.

Numerical Integrators for Dispersion-Managed KdV Equation

In this paper,we consider the numerics of the dispersion-managed Kortewegde Vries(DM-KdV)equation for describingwave propagations in inhomogeneous media.The DM-KdV equation contains a variable dispersion map with discontinuity,which makes the solution non-smooth in time.We formally analyze the convergence order reduction problems of some popular numerical methods including finite difference and time-splitting for solving the DM-KdV equation,where a necessary constraint on the time step has been identified.Then,two exponential-type dispersionmap integrators up to second order accuracy are derived,which are efficiently incorporatedwith the Fourier pseudospectral discretization in space,and they can converge regardless the discontinuity and the step size.Numerical comparisons show the advantage of the proposed methods with the application to solitary wave dynamics and extension to the fast&strong dispersion-management regime.