Lagrange and Multilevel Wavelet-Galerkin with Polynomial Time Basis for Heat Equation

The Wavelet-Galerkin finite element method for solving the one-dimensional heat equation is presented in this work. Two types of basis functions which are the Lagrange and multi-level wavelet bases are employed to derive the full form of matrix system. We consider both linear and quadratic bases in the Galerkin method. Time derivative is approximated by polynomial time basis that provides easily extend the order of approximation in time space. Our numerical results show that the rate of convergences for the linear Lagrange and the linear wavelet bases are the same and in order 2 while the rate of convergences for the quadratic Lagrange and the quadratic wavelet bases are approximately in order 4. It also reveals that the wavelet basis provides an easy treatment to improve numerical resolutions that can be done by increasing just its desired levels in the multilevel construction process.

Authors:

• Watcharakorn Thongchuay
• Puntip Toghaw
• Montri Maleewong

Keywords:

• Galerkin finite element method
• Heat equation
• Lagrange basis function
• Wavelet basis function.

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