On sign-changing solution for a fourth-order asymptotically linear elliptic problem

In this paper we deal with a fourth-order elliptic problem whose nonlinear term is asymptotically linear at both zero and infinity. By using the variational method, we obtain an existence result of sign-changing solutions as well as positive and negative solutions.     

Continuity in two dimensions for a very degenerate elliptic equation

An elliptic equation ∇⋅(F(∇u))=f whose ellipticity strongly degenerates for small values of ∇u (say, F=0 on B(0,1)) is considered. The aim is to prove regularity for F(∇u). The paper proves a continuity result in dimension 2 and presents some applications.     

2D internal flux compatibility equation of the flux Green element method for transient nonlinear potential problems

This article presents the derivation and implementation of the normal directional flux compatibility equation (relationship) at internal nodes when the Green element formulation that consistently provides accurate estimates of the primary variable, and its normal directional derivative (normal flux) is applied in 2D heterogeneous media to steady and transient potential problems. Such a relationship is required to resolve the closure problem due to having fewer integral equations than the number of unknowns at internal nodes. The derivation of the relationship is based on Stokes' theorem, which transforms the contour integral of the normal directional fluxes into a surface integral that is identically zero. The numerical discretization of the compatibility equation is demonstrated with four numerical examples using the six‐node quadratic triangular and the four and eight‐node rectangular elements. The incorporation of triangular elements into the current formulation demonstrates that the internal compatibility equation can be successfully implemented on irregular grids. The direct calculation of the fluxes significantly enhances the accuracy of the formulation, so that high accuracy, exceeding that of the finite element method, is achieved with very coarse spatial discretization. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2010     

A multiresolution space‐time adaptive scheme for the bidomain model in electrocardiology

The bidomain model of electrical activity of myocardial tissue consists of a possibly degenerate parabolic PDE coupled with an elliptic PDE for the transmembrane and extracellular potentials, respectively. This system of two scalar PDEs is supplemented by a time‐dependent ODE modeling the evolution of the gating variable. In the simpler subcase of the monodomain model, the elliptic PDE reduces to an algebraic equation. Since typical solutions of the bidomain and monodomain models exhibit wavefronts with steep gradients, we propose a finite volume scheme enriched by a fully adaptive multiresolution method, whose basic purpose is to concentrate computational effort on zones of strong variation of the solution. Time adaptivity is achieved by two alternative devices, namely locally varying time stepping and a Runge‐Kutta‐Fehlberg‐type adaptive time integration. A series of numerical examples demonstrates that these methods are efficient and sufficiently accurate to simulate the electrical activity in myocardial tissue with affordable effort. In addition, the optimal choice of the threshold for discarding nonsignificant information in the multiresolution representation of the solution is addressed, and the numerical efficiency and accuracy of the method is measured in terms of CPU time speed‐up, memory compression, and errors in different norms. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2010     

A robust optimal preconditioner for the mixed finite element discretization of elliptic optimal control problems

In this paper, we consider the efficient solving of the resulting algebraic system for elliptic optimal control problems with mixed finite element discretization. We propose a block‐diagonal preconditioner for the symmetric and indefinite algebraic system solved with minimum residual method, which is proved to be robust and optimal with respect to both the mesh size and the regularization parameter. The block‐diagonal preconditioner is constructed based on an isomorphism between appropriately chosen solution space and its dual for a general control problem with both state and gradient state observations in the objective functional. Numerical experiments confirm the efficiency of our proposed preconditioner.     

Smoothness of the gradient of weak solutions of degenerate linear equations

Let Q(x) be a nonnegative definite, symmetric matrix such that (Q(x))(1/2) is Lipschitz continuous. Given a real-valued function b(x) and a weak solution u(x) of div(Q▽u) = b, we find sufficient conditions in order that Q(1/2)▽u has some first order smoothness. Specifically, if Ω is a bounded open set in R~n, we study when the components of Q(1/2)▽u belong to the first order Sobolev space W_Q~(1,2)(Ω)defined by Sawyer and Wheeden. Alternately, we study when each of n first order Lipschitz vector field derivatives X_iu has some first order smoothness if u is a weak solution in Ω of ∑_(i=1)~n X′_iX_(iu) + b = 0.We do not assume that {X_i} is a Hormander collection of vector fields in Ω. The results signal ones for more general equations.     

A weak Galerkin finite element method for the second order elliptic problems with mixed boundary conditions

In this paper, a weak Galerkin finite element method is proposed and analyzed for the second-order elliptic equation with mixed boundary conditions. Optimal order error estimates are established in both discrete $H^1$ norm and the standard $L^2$ norm for the corresponding WG approximations. The numerical experiments are presented to verify the efficiency of the method.     

Elliptic function solutions for some nonlinear pdes in mathematical physics

In this work, we have constructed various types of soliton solutions of the generalized regularized long wave and generalized nonlinear Klein-Gordon equations by the using of the extended trial equation method. Some of the obtained exact traveling wave solutions to these nonlinear problems are the rational function, 1-soliton, singular, the elliptic integral functions $F, E, \Pi$ and the Jacobi elliptic function sn solutions. Also, all of the solutions are compared with the exact solutions in literature, and it is seen that some of the solutions computed in this paper are new wave solutions.     

Existence of nonzero positive solutions of systems of second order

Existence of nonzero positive solutions  of systems of second order elliptic boundary value problems under sublinear conditions is obtained. The methodology is to establish a new result on existence of nonzero positive solutions of a fixed point equation in real Banach spaces by using the well-known theory of fixed point index for compact maps defined on cones, where the fixed point equation involves composition of a compact linear operator and a continuous nonlinear map. The conditions imposed on the nonlinear maps involve the spectral radii of the compact linear operators. Moreover, the nonlinear maps are not required to be increasing in  ordered Banach spaces.