A new numerical approximation method for a class of Gaussian random fields on compact connected oriented Riemannian manifolds is introduced. This class of random fields is characterized by the Laplace–Beltrami operator on the manifold. A Galerkin approximation is combined with a polynomial approximation using Chebyshev series. This so-called Galerkin–Chebyshev approximation scheme yields efficient and generic sampling algorithms for Gaussian random fields on manifolds. Strong and weak orders of convergence for the Galerkin approximation and strong convergence orders for the Galerkin–Chebyshev approximation are shown and confirmed through numerical experiments.

Forsythe formulated a conjecture about the asymptotic behavior of the restarted conjugate gradient method in 1968. We translate several of his results into modern terms, and pose an analogous version of the conjecture (originally formulated only for symmetric positive definite matrices) for symmetric and nonsymmetric matrices. Our version of the conjecture uses a two-sided or cross iteration with the given matrix and its transpose, which is based on the projection process used in the Arnoldi (or for symmetric matrices the Lanczos) algorithm. We prove several new results about the limiting behavior of this iteration, but the conjecture still remains largely open. We hope that our paper motivates further research that eventually leads to a proof of the conjecture.

We consider entropy conservative and dissipative discretizations of nonlinear conservation laws with implicit time discretizations and investigate the influence of iterative methods used to solve the arising nonlinear equations. We show that Newton’s method can turn an entropy dissipative scheme into an anti-dissipative one, even when the iteration error is smaller than the time integration error. We explore several remedies, of which the most performant is a relaxation technique, originally designed to fix entropy errors in time integration methods. Thus, relaxation works well in consort with iterative solvers, provided that the iteration errors are on the order of the time integration method. To corroborate our findings, we consider Burgers’ equation and nonlinear dispersive wave equations. We find that entropy conservation results in more accurate numerical solutions than non-conservative schemes, even when the tolerance is an order of magnitude larger.

Line-implicit preconditioners are well known in computational fluid dynamics (CFD) solvers and are an essential component to handle meshes with cells of very high aspect ratio (> 1000:1). Such anisotropic cells are commonly used to resolve steep gradients in the boundary layer of a turbulent flow with high Reynolds number. To date, this technique has rarely been used to solve other partial differential equations. We show that the advantages of such preconditioners do not depend on the partial differential equation or discretization used, but also apply to other problems like a node-based mesh deformation with linear elasticity on such meshes. We show the influence of the selection of these lines, and present a new algorithm for identifying lines for line-implicit preconditioners. This new algorithm makes better use of parallel processors and leads to more homogeneous lines. Finally, we see that using the same line-implicit preconditioner, but the new line identification algorithm, even leads to faster convergence for the mesh deformation problem based on linear elasticity.