A hybrid method using wavelets for the numerical solution of boundary value problems on the interval

Vampa, Victoria; Martín, María Teresa; Serrano, Eduardo

doi:10.1016/j.amc.2010.08.068

Cited by 11 publications

(14 citation statements)

References 13 publications

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“…These drawbacks were analyzed in recent articles [14,15,16] and motivated the development of the Modified Galerkin (MG) method, which combines variational equations with a collocation scheme using both spaces V I j and V I j to construct an algebraic system to obtain u j as follows:…”

Section: Modified Galerkin Methods For a Boundary Value Problemmentioning

confidence: 99%

“…It should be noted that the matrix of equation (8) has a Toeplitz structure and that the final algebraic system that corresponds to the MG method has a band matrix, as sparse as in the case of considering interior basis functions only, but different at the top and at the bottom. Consequently, numerical solutions can be computed efficiently [14,15]. To calculate the matrix elements in equation (8) the following convolution properties of B-spline scaling functions were used [14]:…”

Section: Modified Galerkin Methods For a Boundary Value Problemmentioning

confidence: 99%

“…Consequently, numerical solutions can be computed efficiently [14,15]. To calculate the matrix elements in equation (8) the following convolution properties of B-spline scaling functions were used [14]:…”

Section: Modified Galerkin Methods For a Boundary Value Problemmentioning

confidence: 99%

“…The organization of this paper is as follows: In Section 2, a semi-implicit scheme to advance in time is presented to solve parabolic equations and we described the AWG method to solve boundary value problems: how to define an MRA on the interval [4], the Modified Galerkin method [14] to obtain the approximation at an initial scale in terms of scaling functions and how wavelets are used to increase the scale efficiently [16]. Finally, an error estimation is given.…”

Section: Introductionmentioning

confidence: 99%

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An adaptive Wavelet-Galerkin method for parabolic partial differentia equations

Vampa

Martín

2015

RMTA

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In this paper an Adaptive Wavelet-Galerkin method for the solution of parabolic partial differential equations modeling physical problems with different spatial and temporal scales is developed. A semi-implicit time difference scheme is applied and B-spline multiresolution structure on the interval is used. As in many cases these solutions are known to present localized sharp gradients, local error estimators are designed and an efficient adaptive strategy to choose the appropriate scale for each time is developed. Finally, experiments were performed to illustrate the applicability and efficiency of the proposed method.En este trabajo se desarrolla un método Wavelet-Galerkin Adaptativo para la resolución de ecuaciones diferenciales parabólicas que modelan problemas físicos, con diferentes escalas en el espacio y en el tiempo. Se utiliza un esquema semi-implícito en diferencias temporales y la estructura multirresolución de las B-splines sobre intervalo.Como es sabido que en muchos casos las soluciones presentan gradientes localmente altos, se han diseñado estimadores locales de error y una estrategia adaptativa eficiente para elegir la escala apropiada en cada tiempo. Finalmente, se realizaron experimentos que ilustran la aplicabilidad y la eficiencia del método propuesto.

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Section: Modified Galerkin Methods For a Boundary Value Problemmentioning

confidence: 99%

Section: Modified Galerkin Methods For a Boundary Value Problemmentioning

confidence: 99%

Section: Modified Galerkin Methods For a Boundary Value Problemmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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An adaptive Wavelet-Galerkin method for parabolic partial differentia equations

Vampa

Martín

2015

RMTA

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“…The wavelet method can be viewed as a method in which the approximating function is defined by a multiresolution technique based on scaling or wavelet functions, similar to those used in signal and image processing. With its desirable characteristics, such as multiresolution properties and various basis functions for structural analysis, wavelet method is well argued by many researchers not only in numerical analysis domains [26][27][28] but also in structural analysis fields [29][30][31]. As a kind of wavelet, BSWI basis has the good characteristics of compact support, smoothness, and symmetry besides the multiresolution analysis.…”

Section: Introductionmentioning

confidence: 99%

A Stochastic Wavelet Finite Element Method for 1D and 2D Structures Analysis

Zhang

Chen

Yang

et al. 2014

Shock and Vibration

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A stochastic finite element method based on B-spline wavelet on the interval (BSWI-SFEM) is presented for static analysis of 1D and 2D structures in this paper. Instead of conventional polynomial interpolation, the scaling functions of BSWI are employed to construct the displacement field. By means of virtual work principle and BSWI, the wavelet finite elements of beam, plate, and plane rigid frame are obtained. Combining the Monte Carlo method and the constructed BSWI elements together, the BSWI-SFEM is formulated. The constructed BSWI-SFEM can deal with the problems of structural response uncertainty caused by the variability of the material properties, static load amplitudes, and so on. Taking the widely used Timoshenko beam, the Mindlin plate, and the plane rigid frame as examples, numerical results have demonstrated that the proposed method can give a higher accuracy and a better constringency than the conventional stochastic finite element methods.

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