Large-scale 3D martensitic microstructure evolution problems are studied using a finite-element discretization of finite-strain phase-field model. The model admits an arbitrary crystallography transformation and elastic anisotropy the phases, incorporates Hencky-type elasticity, penalty-regularized double-obstacle potential, viscous dissipation. is performed in Firedrake relies on PETSc solver ...

GMRES(k) is widely used for solving nonsymmetric linear systems. However, it is inadequate either when it converges only for k close to the problem size or when numerical error in the modified Gram–Schmidt process used in the GMRES orthogonalization phase dramatically affects the algorithm performance. An adaptive version of GMRES(k) which tunes the restart value k based on criteria estimating ...

In this paper we derive a posteriori error estimates for the compositional model of multiphase Darcy flow in porous media, consisting of a system of strongly coupled nonlinear unsteady partial differential and algebraic equations. We show how to control the dual norm of the residual augmented by a nonconformity evaluation term by fully computable estimators. We then decompose the estimators int...

A second-order accurate kernel-free boundary integral method is presented for Stokes and Navier value problems on three-dimensional irregular domains. It solves equations in the framework of equations, whose corresponding discrete forms are well-conditioned solved by GMRES method. notable feature this approach that or volume integrals encountered BIEs indirectly evaluated a Cartesian grid-based...

Krylov methods provide a fast and highly parallel numerical tool for the iterative solution of many large-scale sparse linear systems. To large extent, performance practical realizations these is constrained by communication bandwidth in current computer architectures, motivating investigation sophisticated techniques to avoid, reduce, and/or hide message-passing costs (in distributed platforms...

We present a treecode-accelerated boundary integral (TABI) solver for electrostatics of solvated biomolecules described by the linear Poisson-Boltzmann equation. The method employs a wellconditioned boundary integral formulation for the electrostatic potential and its normal derivative on the molecular surface. The surface is triangulated and the integral equations are discretized by centroid c...

We consider the computation of a few eigenvectors and corresponding eigen-values of a large sparse nonsymmetric matrix. In order to compute eigenvaluesin an isolated cluster around a given shift we apply shift-and-invert Arnoldi’smethod with and without implicit restarts. For the inner iterations we useGMRES as the iterative solver. The costs of the inexact solves are measured<l...

In this paper,wepresent a parallel higher-order boundary integralmethod to solve the linear Poisson–Boltzmann (PB) equation. In our method, a well-posed boundary integral formulation is used to ensure the fast convergence of Krylov subspace linear solver such as GMRES. The molecular surfaces are first discretized with flat triangles and then converted to curved triangleswith the assistance of n...

Almost singular linear systems arise in discrete ill-posed problems. Either because of the intrinsic structure of the problem or because of preconditioning, the spectrum of the coefficient matrix is often characterized by a sizable gap between a large group of numerically zero eigenvalues and the rest of the spectrum. Correspondingly, the right-hand side has leading eigencomponents associated w...