Abstract
We propose a numerical validation of a probabilistic approach applied to estimate the relative accuracy between two Lagrange finite elements Pk and Pm,(k < m). In particular, we show practical cases where finite element Pk gives more accurate results than finite element Pm. This illustrates the theoretical probabilistic framework we recently derived in order to evaluate the actual accuracy. This also highlights the importance of the extra caution required when comparing two numerical methods, since the classical results of error estimates concerns only the asymptotic convergence rate.
Highlights
Finite element methods and among them, error estimates play a significant role in the development of numerical methods
We have considered the approximation error as a random variable, and we have evaluated the relative accuracy between two Lagrange finite elements with the help of a probabilistic approach
In [11, 12], we introduced a probabilistic approach that provides a coherent framework for modeling uncertainties in finite element approximations: such uncertainties may come from the way the meshes are created by computer algorithms, leading to a partial non-control of the mesh, even for a given maximum mesh size
Summary
Finite element methods and among them, error estimates play a significant role in the development of numerical methods. We are concerned with a priori error estimates, that aim to find upper bounds for the error between the exact solution u and its finite element approximation uh. These estimates describe how the finite element error u − uh , for a given norm, goes to 0 with mesh size h (i.e. the largest diameter of the elements in a given mesh). In previous papers [11,12], we investigated the error resulting from a partial non-control of the mesh size For this purpose, we have considered the approximation error as a random variable, and we have evaluated the relative accuracy between two Lagrange finite elements with the help of a probabilistic approach.
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