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

doi:10.1007/s10483-022-2859-5

Abstract

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.

Key words: wavelet multiresolution interpolation, transcendental nonlinearity, localized steep gradient, singularly perturbed boundary value problem, Troesch's problem

2010 MSC Number:

65L11
34B15

Xiaojing LIU, Youhe ZHOU, Jizeng WANG. Wavelet multiresolution interpolation Galerkin method for nonlinear boundary value problems with localized steep gradients. Applied Mathematics and Mechanics (English Edition), 2022, 43(6): 863-882.

TrendMD

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