Mesh Independence Principle Research Articles

A Multilevel Mesh Independence Principle for the Navier–Stokes Equations

Multilevel, finite element discretization methods for the Navier–Stokes equations are considered. In contrast to usual multilevel methods, a superlinear scaling of the consecutive meshwidths $h_J + 1 = \mathcal {O}(h_j^{\alpha (j)} )$ is used. On the coarsest mesh the discretized system is solved, which is small and nonlinear. On each subsequent mesh only one or two Newton correction steps are performed, so that only one or two larger, linear systems are solved. The scalings of the meshwidths that lead to optimal accuracy of the approximate solution in both the $H^1 $- and $L^2 $-norm are investigated.An error analysis of independent interest is also presented for the basic finite element method using the conservation form of the nonlinear term. This formulation leads to a nonlinear system whose nonlinearity is easier to resolve than the systems arising when either the convective or explicitly skew-symmetric corms are used for the nonlinear term.

On a mesh-independence principle for operator equations and the secant method

On a mesh-independence principle for operator equations and the secant method

A mesh-independence principle for nonlinear operator equations and their discretizations under mild differentiability conditions

In this note we extend the validity of the mesh-independence principle for nonlinear operator equations and their discretizations to include operators whose derivatives are only Holder continuous.

A mesh independence principle for nonlinear equations using newton's method ano nonlinear projections.

We consider the nonlinear operator equation in a Banach space. We make use of nonlinear projections on finite dimensional spaces to produce the finite dimensional discretization of the nonlinear equation. Using Newton's method we then prove the mesh-independence principle for this problem. Our results cover and extend previous results involving linear projections on finite dimensional spaces.

Application of the Mesh Independence Principle to Mesh Refinement Strategies

The Mesh Independence Principle (MIP) for a Newton iteration applied to an operator equation and its discretization for different “stepsizes” h states for a fixed starting point the number of iterations of Newton’s method to obtain a fixed tolerance is eventually independent of h. This MIP is used to formulate an efficient strategy for the solution of nonlinear equations amounting to the work equivalent of two or three Newton iterates for the smallest h. Examples are given.

A Mesh-Independence Principle for Operator Equations and Their Discretizations

The mesh-independence principle asserts that, when Newton’s method is applied to a nonlinear equation between some Banach spaces as well as to some finite-dimensional discretization of that equation, then the behavior of the discretized process is asymptotically the same as that for the original iteration and, as a consequence, the number of steps required by the two processes to converge to within a given tolerance is essentially the same. So far this result has been proved only for certain classes of boundary value problems. In this paper a proof is presented for a general class of operator equations and discretizations. It covers the earlier results and extends them well beyond the cases that have been considered before.

A general mesh independence principle for Newton's method applied to second order boundary value problems

Recent work has established that for certain classes of nonlinear boundary value problems, the number of Newton iterations applied to the related standard discrete problem for a given tolerance is independent of the mesh size when the mesh is sufficiently fine. This paper develops an extension of the mesh independence principle by relaxing the assumption on the differential equation, its boundary conditions, and the related difference approximation.