Abstract
In finite element methods, the accuracy of the solution cannot increase indefinitely since the round-off error related to limited computer precision increases when the number of degrees of freedom (DoFs) is large enough. Because a priori information of the highest attainable accuracy is of great interest, we construct an innovative method to obtain the highest attainable accuracy given the order of the elements. In this method, the truncation error is extrapolated when it converges at the asymptotic rate, and the bound of the round-off error follows from a generically valid error estimate, obtained and validated through extensive numerical experiments. The highest attainable accuracy is obtained by minimizing the sum of these two types of errors. We validate this method using a one-dimensional Helmholtz equation in space. It shows that the highest attainable accuracy can be accurately predicted, and the CPU time required is much smaller compared with that using successive grid refinement.
Highlights
Many problems in engineering sciences and industry are modelled mathematically by initial–boundary value problems comprising systems of coupled, nonlinear partial and/or ordinary differential equations
The accuracy of the numerically obtained solution is influenced by many sources of errors [3]: firstly, modelling errors in the set-up of the models, such as the simplification of realistic domains and governing equations and the approximation of initial and boundary conditions; truncation errors due to the discretization of the computational domain and the use of basis functions for the function spaces defined on it; round-off errors due to the adoption of finite-precision computer arithmetics, rather than exact arithmetics; iteration errors resulting from the artificially controlled tolerance of iterative solvers
In contrast to the brute-force approach, which uses successive h-refinements, this approach uses only a few coarse grid refinements to reach the region of asymptotic convergence
Summary
Many problems in engineering sciences and industry are modelled mathematically by initial–boundary value problems comprising systems of coupled, nonlinear partial and/or ordinary differential equations. These problems often consider complex geometries, with initial and/or boundary conditions that depend on measured data [1]. For many problems of practical interest, analytical or semi-analytical solutions are not available, and one has to resort to numerical solution methods, such as the finite difference, finite volume, and finite element methods The latter will be adopted throughout this paper and applied to one-dimensional boundary value problems. The accuracy of the numerically obtained solution is influenced by many sources of errors [3]: firstly, modelling errors in the set-up of the models, such as the simplification of realistic domains and governing equations and the approximation of initial and boundary conditions; truncation errors due to the discretization of the computational domain and the use of basis functions for the function spaces defined on it; round-off errors due to the adoption of finite-precision computer arithmetics, rather than exact arithmetics; iteration errors resulting from the artificially controlled tolerance of iterative solvers
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