Simple Test Functions in Meshless Local Petrov-Galerkin Methods

Quasi-optimal and pressure-robust discretizations of the Stokes equations by new augmented Lagrangian formulations

IMA Journal of Numerical Analysis ◽

10.1093/imanum/drz044 ◽

2019 ◽

Vol 40 (4) ◽

pp. 2553-2583

Author(s):

Christian Kreuzer ◽

Pietro Zanotti

Keyword(s):

Finite Element ◽

Linear Operator ◽

Augmented Lagrangian ◽

Discontinuous Galerkin Methods ◽

Stokes Equations ◽

Galerkin Methods ◽

Test Functions ◽

Lagrangian Formulations ◽

Stable Pairs ◽

Augmented Lagrangian Formulation

Abstract We approximate the solution of the stationary Stokes equations with various conforming and nonconforming inf-sup stable pairs of finite element spaces on simplicial meshes. Based on each pair, we design a discretization that is quasi-optimal and pressure-robust, in the sense that the velocity $H^1$-error is proportional to the best velocity $H^1$-error. This shows that such a property can be achieved without using conforming and divergence-free pairs. We also bound the pressure $L^2$-error, only in terms of the best velocity $H^1$-error and the best pressure $L^2$-error. Our construction can be summarized as follows. First, a linear operator acts on discrete velocity test functions, before the application of the load functional, and maps the discrete kernel into the analytical one. Second, in order to enforce consistency, we employ a new augmented Lagrangian formulation, inspired by discontinuous Galerkin methods.

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A class of discontinuous Petrov-Galerkin methods. II. Optimal test functions

Numerical Methods for Partial Differential Equations ◽

10.1002/num.20640 ◽

2010 ◽

Vol 27 (1) ◽

pp. 70-105 ◽

Cited By ~ 117

Author(s):

L. Demkowicz ◽

J. Gopalakrishnan

Keyword(s):

Galerkin Methods ◽

Test Functions ◽

Optimal Test

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Predictive Models for the Breeder Genetic Algorithm I. Continuous Parameter Optimization

Evolutionary Computation ◽

10.1162/evco.1993.1.1.25 ◽

1993 ◽

Vol 1 (1) ◽

pp. 25-49 ◽

Cited By ~ 766

Author(s):

Heinz Mühlenbein ◽

Dirk Schlierkamp-Voosen

Keyword(s):

Genetic Algorithm ◽

Predictive Model ◽

Parameter Optimization ◽

Quantitative Genetics ◽

Artificial Selection ◽

Predictive Models ◽

Simple Test ◽

Test Functions ◽

Continuous Parameter ◽

Numerical Performance

In this paper a new genetic algorithm called the Breeder Genetic Algorithm (BGA) is introduced. The BGA is based on artificial selection similar to that used by human breeders. A predictive model for the BGA is presented that is derived from quantitative genetics. The model is used to predict the behavior of the BGA for simple test functions. Different mutation schemes are compared by computing the expected progress to the solution. The numerical performance of the BGA is demonstrated on a test suite of multimodal functions. The number of function evaluations needed to locate the optimum scales only as n ln(n) where n is the number of parameters. Results up to n = 1000 are reported.

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