Imposing boundary and interface conditions in multi-resolution wavelet Galerkin method for numerical solution of Helmholtz problems

• Development of a new multi-resolution wavelet Galerkin method for solving boundary value problems.
• New form of jump functions proposed in order to apply general boundary conditions.
• Two different polynomial and slope jump functions are examined in numerical analysis of boundary value problems.
• Wavelet Galerkin solution of Laplace and Helmholtz equations with different boundary conditions.

In this paper, we propose a new formulation for numerical solution of boundary value problems using mesh-free multi-resolution wavelet Galerkin (WG) method. A difficulty with the WG method is imposing boundary condition constraints. Single scale wavelet basis functions together with a jump function approach using cubic polynomial functions have been used in solving some boundary problems in the literature. Ramp and cubic polynomial functions have been also used in the jump function approach described in element-free Galerkin context, and it is proven ramp functions exactly satisfy the desirable property for the derivative of jump, but the cubic polynomial functions do not. In this paper, a multi-resolution WG method is employed, and both ramp and cubic polynomial jump functions are applied. Obtained results in one dimension are compared with analytical solutions. Results obtained with ramp jump functions in two dimensions are also reported. The results obtained with the WG method manifest boundary and interface conditions are accurately imposed when the ramp jump functions are applied. Numerical results are in good agreement with accurate solutions.