Monotonicity recovering and accuracy preserving optimization methods for postprocessing finite element solutions

Abstract

We suggest here a least-change correction to available finite element (FE) solution. This postprocessing procedure is aimed at recovering the monotonicity and some other important properties that may not be exhibited by the FE solution. It is based on solving a monotonic regression problem with some extra constraints. One of them is a linear equality-type constraint which models the conservativity requirement. The other ones are box-type constraints, and they originate from the discrete maximum principle. The resulting postprocessing problem is a large scale quadratic optimization problem. It is proved that the postprocessed FE solution preserves the accuracy of the discrete FE approximation.We introduce an algorithm for solving the postprocessing problem. It can be viewed as a dual ascent method based on the Lagrangian relaxation of the equality constraint. We justify theoretically its correctness. Its efficiency is demonstrated by the presented results of numerical experiments.

A Dual Active-Set Algorithm for Regularized Monotonic Regression

“…As it was shown above, the SPAV algorithm generates at each iteration an active set S such that x(S) and λ(S) satisfy conditions (9), (11), and (12) of all the KKT conditions, but not (10). Since it aims at attaining primal feasibility while maintaining dual feasibility and complementary slackness, the SPAV can be viewed as a dual activeset algorithm, even though the Lagrange multipliers are not calculated.…”

“…Examples are found in such areas as operations research [3], genetics [4], environmental science [5], meteorology [6], psychology [7], and many others. One can find very large-scale MR problems, e.g., in machine learning [8][9][10] and computer simulations [11].…”

On enforcing maximum principles and achieving element-wise species balance for advection–diffusion–reaction equations under the finite element method

“…Approach #1: Clipping/cut-off methods. There are various post-processing procedures such as clipping/cut-off methods [22,34] to ensure that a certain numerical formulation satisfies the non-negative constraint. The key idea of these methods is to simply chop-off the negative values in a numerical solution.…”

A monotone nonlinear finite volume method for approximating diffusion operators on general meshes

“…However, it is known that none of these methods does satisfy the discrete maximum principle except in the particular case of isotropic diffusion equations discretized on some particular meshes like rectangle meshes or, more generally, Delaunay–Voronoi meshes and admissible meshes . When the approximated solution must be positive (for example, temperature, energy, concentration, and so on) this drawback can become embarrassing and can require repair techniques or monotony constraints , in order to enforce the approximated solution to be positive.…”

“…MONOTONE NONLINEAR FINITE VOLUME METHOD 497 meshes like rectangle meshes or, more generally, Delaunay-Voronoi meshes [26] and admissible meshes [27]. When the approximated solution must be positive (for example, temperature, energy, concentration, and so on) this drawback can become embarrassing and can require repair techniques [28][29][30][31] or monotony constraints [32,33], in order to enforce the approximated solution to be positive.…”