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52
Edge stabilization for Galerkin approximations of convectiondiffusionreaction problems
 Comp. Methods Appl. Mech. Engrg
"... Abstract. We analyze a nonlinear shockcapturing scheme for H 1conforming, piecewiseaffine finite element approximations of linear elliptic problems. The meshes are assumed to satisfy two standard conditions: a local quasiuniformity property and the Xu–Zikatanov condition ensuring that the stiffne ..."
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Cited by 75 (20 self)
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Abstract. We analyze a nonlinear shockcapturing scheme for H 1conforming, piecewiseaffine finite element approximations of linear elliptic problems. The meshes are assumed to satisfy two standard conditions: a local quasiuniformity property and the Xu–Zikatanov condition ensuring that the stiffness matrix associated with the Poisson equation is an Mmatrix. A discrete maximum principle is rigorously established in any space dimension for convectiondiffusionreaction problems. We prove that the shockcapturing finite element solution converges to that without shockcapturing if the cell Péclet numbers are sufficiently small. Moreover, in the diffusiondominated regime, the difference between the two finite element solutions superconverges with respect to the actual approximation error. Numerical experiments on test problems with stiff layers confirm the sharpness of the a priori error estimates. 1.
Mesh Generation
 HANDBOOK OF COMPUTATIONAL GEOMETRY. ELSEVIER SCIENCE
, 2000
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The finite element approximation of the nonlinear poissonboltzmann equation
 SIAM Journal on Numerical Analysis
"... ABSTRACT. A widely used electrostatics model in the biomolecular modeling community, the nonlinear Poisson–Boltzmann equation, along with its finite element approximation, are analyzed in this paper. A regularized Poisson–Boltzmann equation is introduced as an auxiliary problem, making it possible t ..."
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Cited by 31 (12 self)
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ABSTRACT. A widely used electrostatics model in the biomolecular modeling community, the nonlinear Poisson–Boltzmann equation, along with its finite element approximation, are analyzed in this paper. A regularized Poisson–Boltzmann equation is introduced as an auxiliary problem, making it possible to study the original nonlinear equation with delta distribution sources. A priori error estimates for the finite element approximation are obtained for the regularized Poisson–Boltzmann equation based on certain quasiuniform grids in two and three dimensions. Adaptive finite element approximation through local refinement driven by an a posteriori error estimate is shown to converge. The Poisson–Boltzmann equation does not appear to have been previously studied in detail theoretically, and it is hoped that this paper will help provide molecular modelers with a better foundation for their analytical and computational work with the Poisson–Boltzmann equation. Note that this article apparently gives the first rigorous convergence result for a numerical discretization technique for the nonlinear Poisson– Boltzmann equation with delta distribution sources, and it also introduces the first provably convergent adaptive method for the equation. This last result is currently one of only a handful of existing convergence results of this type for nonlinear problems.
Failure of the discrete maximum principle for an elliptic finite element problem
 Math. Comp
"... Abstract. There has been a longstanding question of whether certain mesh restrictions are required for a maximum condition to hold for the discrete equations arising from a finite element approximation of an elliptic problem. This is related to knowing whether the discrete Green’s function is posit ..."
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Cited by 27 (0 self)
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Abstract. There has been a longstanding question of whether certain mesh restrictions are required for a maximum condition to hold for the discrete equations arising from a finite element approximation of an elliptic problem. This is related to knowing whether the discrete Green’s function is positive for triangular meshes allowing sufficiently good approximation of H 1 functions. We study this question for the Poisson problem in two dimensions discretized via the Galerkin method with continuous piecewise linears. We give examples which show that in general the answer is negative, and furthermore we extend the number of cases where it is known to be positive. Our techniques utilize some new results about discrete Green’s functions that are of independent interest. 1.
Unstructured grid optimization for improved monotonicity of discrete solutions of elliptic equations with highly anisotropic coefficients
, 2006
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On the convergence of finite element methods for HamiltonJacobiBellman equations
 SIAM J. Numer. Anal
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On the existence of maximum principles in parabolic finite element equations
 Math. Comp
, 1999
"... Abstract. In 1973, H. Fujii investigated discrete versions of the maximum principle for the model heat equation using piecewise linear finite elements in space. In particular, he showed that the lumped mass method allows a maximum principle when the simplices of the triangulation are acute, and this ..."
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Cited by 7 (1 self)
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Abstract. In 1973, H. Fujii investigated discrete versions of the maximum principle for the model heat equation using piecewise linear finite elements in space. In particular, he showed that the lumped mass method allows a maximum principle when the simplices of the triangulation are acute, and this is known to generalize in two space dimensions to triangulations of Delauney type. In this note we consider more general parabolic equations and first show that a maximum principle cannot hold for the standard spatially semidiscrete problem. We then show that for the lumped mass method the above conditions on the triangulation are essentially sharp. This is in contrast to the elliptic case in which the requirements are weaker. We also study conditions for the solution operator acting on the discrete initial data, with homogeneous lateral boundary conditions, to be a contraction or a positive operator. 1.
MultiResolution Approximate Inverse Preconditioners
 SIAM J. Sci. Comput
, 2001
"... . We introduce a new preconditioner for elliptic PDE's on unstructured meshes. Using a waveletinspired basis we compress the inverse of the matrix, allowing an e#ective sparse approximate inverse by solving the sparsity vs. accuracy conflict. The key issue in this compression is to use second ..."
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Cited by 7 (0 self)
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. We introduce a new preconditioner for elliptic PDE's on unstructured meshes. Using a waveletinspired basis we compress the inverse of the matrix, allowing an e#ective sparse approximate inverse by solving the sparsity vs. accuracy conflict. The key issue in this compression is to use secondgeneration wavelets which can be adapted to the unstructured mesh, the true boundary conditions, and even the PDE coe#cients. We also show how this gives a new perspective on multiresolution algorithms such as multigrid, interpreting the new preconditioner as a variation on nodenested multigrid. In particular, we hope the new preconditioner will combine the best of both worlds: fast convergence when multilevel methods can succeed, but with robust performance for more di#cult problems. The rest of the paper discusses the core issues for the preconditioner: ordering and construction of a factored approximate inverse in the multiresolution basis, robust interpolation on unstructured meshes, automa...
A MATHEMATICAL MODEL FOR THE HARD SPHERE REPULSION IN IONIC SOLUTIONS
, 2010
"... We introduce a mathematical model for finite size (repulsive) effects in ionic solutions. We first introduce an appropriate energy term into the total energy that represents the hard sphere repulsion of ions. The total energy then consists of the entropic energy, electrostatic potential energy, and ..."
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Cited by 7 (6 self)
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We introduce a mathematical model for finite size (repulsive) effects in ionic solutions. We first introduce an appropriate energy term into the total energy that represents the hard sphere repulsion of ions. The total energy then consists of the entropic energy, electrostatic potential energy, and the repulsive potential energy. The energetic variational approach is then used to derive a boundary value problem (the ‘EulerLagrange’ equations) that includes contributions from the repulsive term. The resulting system of partial differential equations is a modification of the PoissonNernstPlanck (PNP) equations widely if not universally used to describe the driftdiffusion of electrons and holes in semiconductors, and the movement of ions in solutions and protein channels. The modified PNP equations include the effects of the finite size of ions that are so important in the concentrated solutions near electrodes, active sites of enzymes, and selectivity filters of proteins. Finally, we do some numerical experiments using finite element methods, and present their results as a verification of the utility of the modified system.
Adaptivity in 3D Image Processing
 COMPUTING AND VISUALIZATION IN SCIENCE
, 2001
"... We present an adaptive numerical scheme for computing the nonlinear partial differential equations arising in 3D image multiscale analysis. The scheme is based on a semiimplicit scale discretization and on an adaptive finite element method in 3Dspace. Successive coarsening of the computational gr ..."
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Cited by 6 (3 self)
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We present an adaptive numerical scheme for computing the nonlinear partial differential equations arising in 3D image multiscale analysis. The scheme is based on a semiimplicit scale discretization and on an adaptive finite element method in 3Dspace. Successive coarsening of the computational grid is used for increasing the efficiency of the numerical procedure. L1 stability of the suggested numerical method is presented and computational results related to 3D nonlinear image filtering are discussed.