Article Contents
Article Contents

# Boundary value problem: Weak solutions induced by fuzzy partitions

• * Corresponding author: Irina Perfilieva

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

• 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.

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

 Citation:

• 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 $3.28\times10^{-6}$ $2.15\times10^{-8}$ $1.44\times10^{-9}$ $2.16\times10^{-10}$

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 $9.20\times10^{-5}$ $1.55\times10^{-6}$ $2.23\times10^{-8}$ $5.54\times10^{-10}$

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 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 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 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 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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