Boundary value problem: Weak solutions induced by fuzzy partitions

Linh Nguyen; Irina Perfilieva; Michal Holčapek

doi:10.3934/dcdsb.2019263

Article Contents

2020, Volume 25, Issue 2: 715-732. Doi: 10.3934/dcdsb.2019263

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Boundary value problem: Weak solutions induced by fuzzy partitions

Institute for Research and Applications of Fuzzy Modelling, NSC IT4Innovations, University of Ostrava, 30. dubna 22,701 03 Ostrava 1, Czech Republic

^* Corresponding author: Irina Perfilieva
^* Corresponding author: Irina Perfilieva

Received: December 31, 2018

Revised: July 31, 2019

Early access: November 2019

Published: February 2020

This research was supported by the Czech Ministry of Education, Youth and Sports, project OP VVV (AI-Met4AI): No. CZ.02.1.01/0.0/0.0/17-049/0008414. Additional support was given by the Grant Agency of the Czech Republic, project 18-06915S

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Abstract

The aim of this paper is to propose a new methodology in the construction of spaces of test functions used in a weak formulation of the Boundary Value Problem. The proposed construction is based on the so called "two dimensional" approach where at first, we select a partition of a domain and second, a dimension of an approximating functional subspace on each partition element. The main advantage consists in the independent selection of the key parameters, aiming at achieving a requested quality of approximation with a reasonable complexity. We give theoretical justification and illustration on examples that confirm our methodology.

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Mathematics Subject Classification: Primary: 34L99, 65N30; Secondary: 65N99, 03E72.

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Table 1. The approximation quality with respect to the degree of polynomials ($ m $) and the number of basic functions ($ N $)

Example 1
$ \bf{m} $ $ \setminus $ $ \bf{N} $	$ 4 $	$ 8 $	$ 12 $	$ 16 $
1	$ 4.24\times10^{-3} $	$ 3.34\times10^{-4} $	$ 8.61\times10^{-5} $	$ 3.39\times10^{-5} $
2	$ 2.34\times10^{-5} $	$ 7.92\times10^{-7} $	$ 1.30\times10^{-7} $	$ 3.70\times10^{-8} $
3	$ 1.44\times10^{-7} $	$ 2.09\times10^{-9} $	$ 2.17\times10^{-10} $	$ 4.55\times10^{-11} $
4	$ 1.10\times10^{-9} $	$ 1.27\times10^{-11} $	$ 1.24\times10^{-11} $	$ 9.41\times10^{-12} $
Example 2
$ \bf{m} $ $ \setminus $ $ \bf{N} $	$ 4 $	$ 8 $	$ 12 $	$ 16 $
1	$ 7.30\times10^{-3} $	$ 1.64\times10^{-3} $	$ 8.83\times10^{-4} $	$ 6.24\times10^{-4} $
2	$ 4.21\times10^{-3} $	$ 1.52\times10^{-3} $	$ 8.80\times10^{-4} $	$ 6.23\times10^{-4} $
3	$ 2.23\times10^{-3} $	$ 7.87\times10^{-4} $	$ 4.86\times10^{-4} $	$ 3.54\times10^{-4} $
4	$ 2.13\times10^{-3} $	$ 7.85\times10^{-4} $	$ 4.75\times10^{-4} $	$ 3.52\times10^{-4} $
Example 3
$ \bf{m} $ $ \setminus $ $ \bf{N} $	$ 4 $	$ 8 $	$ 12 $	$ 16 $
1	$ 1.0308 $	$ 0.7448 $	$ 0.5683 $	$ 0.2451 $
2	$ 0.8802 $	$ 0.5818 $	$ 0.1846 $	$ 9.83\times10^{-2} $
3	$ 0.8501 $	$ 0.3572 $	$ 9.60\times10^{-2} $	$ 2.03\times10^{-2} $
4	$ 0.8338 $	$ 0.2113 $	$ 3.67\times10^{-2} $	$ 7.41\times10^{-3} $
Example 4
$ \bf{m} $ $ \setminus $ $ \bf{N} $	$ 4 $	$ 8 $	$ 12 $	$ 16 $
1	$ 2.07\times10^{-2} $	$ 1.71\times10^{-3} $	$ 4.36\times10^{-4} $	$ 1.72\times10^{-4} $
2	$ 1.40\times10^{-3} $	$ 4.99\times10^{-5} $	$ 8.26\times10^{-6} $	$ 2.37\times10^{-6} $
3	$ 7.52\times10^{-5} $	$ 1.15\times10^{-6} $	$ 1.21\times10^{-7} $	$ 2.49\times10^{-8} $
4	$ 3.28\times10^{-6} $	$ 2.15\times10^{-8} $	$ 1.44\times10^{-9} $	$ 2.16\times10^{-10} $

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Table 2. The approximation quality with respect to the degree of polynomials ($m$) and the number of basic functions ($N$)

Example 3
$\bf{m}$ $\setminus$ $\bf{N}$	$32$	$64$	$128$	$256$
1	$4.40\times10^{-2}$	$5.62\times10^{-3}$	$7.07\times10^{-4}$	$8.78\times10^{-5}$
2	$6.20\times10^{-3}$	$4.13\times10^{-4}$	$2.54\times10^{-5}$	$1.57\times10^{-6}$
3	$9.25\times10^{-4}$	$2.64\times10^{-5}$	$8.08\times10^{-7}$	$2.24\times10^{-8}$
4	$9.20\times10^{-5}$	$1.55\times10^{-6}$	$2.23\times10^{-8}$	$5.54\times10^{-10}$

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Table 3. The comparison of the proposed method (FPP) and the piecewise linear FEM based on the approximation errors and the convergence rates

Example 1
# N	FPP		FEM
# N	Error	Rate	Error	Rate
$ 8 $	$ 1.8\times10^{-3} $	_	$ 2.3\times10^{-2} $	_
$ 16 $	$ 2.2\times10^{-4} $	3.03	$ 4.9\times10^{-3} $	2.23
$ 32 $	$ 2.8\times10^{-5} $	2.97	$ 1.1\times10^{-3} $	2.16
$ 64 $	$ 3.3\times10^{-6} $	3.08	$ 2.8\times10^{-4} $	1.97
$ 128 $	$ 4.5\times10^{-7} $	2.87	$ 6.8\times10^{-5} $	2.04
$ 256 $	$ 5.1\times10^{-8} $	3.14	$ 1.7\times10^{-5} $	2.00
Example 2
# N	FPP		FEM
# N	Error	Rate	Error	Rate
$ 8 $	$ 7.3\times10^{-3} $	_	$ 2.2\times10^{-2} $	_
$ 16 $	$ 1.6\times10^{-3} $	2.19	$ 5.4\times10^{-3} $	2.03
$ 32 $	$ 6.2\times10^{-4} $	1.37	$ 1.7\times10^{-3} $	1.67
$ 64 $	$ 3.0\times10^{-4} $	1.05	$ 6.5\times10^{-4} $	1.39
$ 128 $	$ 1.5\times10^{-4} $	1.00	$ 2.9\times10^{-4} $	1.16
$ 256 $	$ 8.6\times10^{-5} $	8.03	$ 1.4\times10^{-4} $	1.05
Example 3
# N	FPP		FEM
# N	Error	Rate	Error	Rate
$ 8 $	$ 0.9785 $	_	$ 1.0055 $	_
$ 16 $	$ 0.6878 $	5.09	$ 0.8559 $	2.32
$ 32 $	$ 0.2447 $	1.49	$ 0.2764 $	1.63
$ 64 $	$ 3.9\times10^{-2} $	2.65	$ 7.1\times10^{-2} $	1.96
$ 128 $	$ 5.4\times10^{-3} $	2.85	$ 1.8\times10^{-2} $	1.98
$ 256 $	$ 6.9\times10^{-4} $	2.97	$ 4.4\times10^{-3} $	2.03
Example 4
# N	FPP		FEM
# N	Error	Rate	Error	Rate
$ 8 $	$ 1.1\times10^{-2} $	_	$ 4.9\times10^{-2} $	_
$ 16 $	$ 1.7\times10^{-3} $	2.69	$ 1.1\times10^{-2} $	2.16
$ 32 $	$ 1.7\times10^{-4} $	3.32	$ 2.6\times10^{-3} $	2.08
$ 64 $	$ 1.9\times10^{-5} $	3.16	$ 6.2\times10^{-4} $	2.07
$ 128 $	$ 2.4\times10^{-6} $	2.98	$ 1.5\times10^{-4} $	2.05
$ 256 $	$ 3.1\times10^{-7} $	2.95	$ 3.8\times10^{-5} $	1.98

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