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Wavelet multiresolution interpolation Galerkin method for nonlinear boundary value problems with localized steep gradients

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Abstract

The wavelet multiresolution interpolation for continuous functions defined on a finite interval is developed in this study by using a simple alternative of transformation matrix. The wavelet multiresolution interpolation Galerkin method that applies this interpolation to represent the unknown function and nonlinear terms independently is proposed to solve the boundary value problems with the mixed Dirichlet-Robin boundary conditions and various nonlinearities, including transcendental ones, in which the discretization process is as simple as that in solving linear problems, and only common two-term connection coefficients are needed. All matrices are independent of unknown node values and lead to high efficiency in the calculation of the residual and Jacobian matrices needed in Newton’s method, which does not require numerical integration in the resulting nonlinear discrete system. The validity of the proposed method is examined through several nonlinear problems with interior or boundary layers. The results demonstrate that the proposed wavelet method shows excellent accuracy and stability against nonuniform grids, and high resolution of localized steep gradients can be achieved by using local refined multiresolution grids. In addition, Newton’s method converges rapidly in solving the nonlinear discrete system created by the proposed wavelet method, including the initial guess far from real solutions.

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Authors and Affiliations

  1. Key Laboratory of Mechanics on Disaster and Environment in Western China (Lanzhou University), the Ministry of Education, College of Civil Engineering and Mechanics, Lanzhou University, Lanzhou, 730000, China

    Xiaojing Liu, Youhe Zhou & Jizeng Wang

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  1. Xiaojing Liu
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  2. Youhe Zhou
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  3. Jizeng Wang
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Correspondence to Jizeng Wang.

Additional information

Citation: LIU, X. J., ZHOU, Y. H., and WANG, J. Z. Wavelet multiresolution interpolation Galerkin method for nonlinear boundary value problems with localized steep gradients. Applied Mathematics and Mechanics (English Edition), 43(6), 863–882 (2022) https://doi.org/10.1007/s10483-022-2859-5

Project supported by the National Natural Science Foundation of China (Nos. 12172154 and 11925204), the 111 Project of China (No. B14044), and the National Key Project of China (No. GJXM92579)

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Liu, X., Zhou, Y. & Wang, J. Wavelet multiresolution interpolation Galerkin method for nonlinear boundary value problems with localized steep gradients. Appl. Math. Mech.-Engl. Ed. 43, 863–882 (2022). https://doi.org/10.1007/s10483-022-2859-5

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  • DOI: https://doi.org/10.1007/s10483-022-2859-5

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