This paper gives a solution to an open problem concerning the performance of various multilevel preconditioners for the linear finite element approximation of second-order elliptic boundary value problems with strongly discontinuous coefficients. By analyzing the eigenvalue distribution of the BPX preconditioner and multigrid V-cycle preconditioner, we prove that only a small number of eigenvalues may deteriorate with respect to the discontinuous jump or meshsize, and we prove that all the other eigenvalues are bounded below and above nearly uniformly with respect to the jump and meshsize. As a result, we prove that the convergence rate of the preconditioned conjugate gradient methods is uniform with respect to the large jump and meshsize. We also present some numerical experiments to demonstrate the theoretical results.

We consider the design of an effective and reliable adaptive finite element method (AFEM) for the nonlinear Poisson-Boltzmann equation (PBE). We first examine the two-term regularization technique for the continuous problem recently proposed by Chen, Holst, and Xu based on the removal of the singular electrostatic potential inside biomolecules; this technique made possible the development of the first complete solution and approximation theory for the PoissonBoltzmann equation, the first provably convergent discretization, and also allowed for the development of a provably convergent AFEM. However, in practical implementation, this twoterm regularization exhibits numerical instability. Therefore, we examine a variation of this regularization technique which can be shown to be less susceptible to such instability. We establish a priori estimates and other basic results for the continuous regularized problem, as well as for Galerkin finite element approximations. We show that the new approach produces regularized continuous and discrete problems with the same mathematical advantages of the original regularization. We then design an AFEM scheme for the new regularized problem, and show that the resulting AFEM scheme is accurate and reliable, by proving a contraction result for the error. This result, which is one of the first results of this type for nonlinear elliptic problems, is based on using continuous and discrete a priori L ∞ estimates to establish quasi-orthogonality. To provide a high-quality geometric model as input to the AFEM algorithm, we also describe a class of feature-preserving adaptive mesh generation algorithms designed specifically for constructing meshes of biomolecular structures, based on the intrinsic local structure tensor of the molecular surface. All of the algorithms described in the article are implemented in the Finite Element Toolkit (FETK), developed and maintained at UCSD. The stability advantages of the new regularization scheme are demonstrated with FETK through comparisons with the original regularization approach for a model problem. The convergence and accuracy of the overall AFEM algorithm is also illustrated by numerical approximation of electrostatic solvation energy for an insulin protein.

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Abstract. We introduce and analyze two-level and multi-level preconditioners for a family of Interior Penalty (IP) discontinuous Galerkin (DG) discretizations of second order elliptic problems with large jumps in the diffusion coefficient. Our approach to IPDG-type methods is based on a splitting of the DG space into two components that are orthogonal in the energy inner product naturally induced by the methods. As a result, the methods and their analysis depend in a crucial way on the diffusion coefficient of the problem. The analysis of the proposed preconditioners is presented for both symmetric and non-symmetric IP schemes; dealing simultaneously with the jump in the diffusion coefficient and the non-nested character of the relevant discrete spaces presents extra difficulties in the analysis which precludes a simple extension of existing results. However, we are able to establish robustness (with respect to the diffusion coefficient) and nearly-optimality (up to a logarithmic term depending on the mesh size) for both two-level and BPX-type preconditioners. Following the analysis, we present a sequence of detailed numerical results which verify the theory and illustrate the performance of the methods. The paper includes an Appendix with a collection of proofs of several technical results required for the analysis.