Fast computation of elastic and hydrodynamic potentials using approximate approximations

Abstract

We propose fast cubature formulas for the elastic and hydrodynamic potentials based on the approximate approximation of the densities with Gaussian and related functions. For densities with separated representation, we derive a tensor product representation of the integral operator which admits efficient cubature procedures. We obtain high order approximations up to a small saturation error, which is negligible in computations. Results of numerical experiments which show approximation order $$\mathscr {O}(h^{2M})$$ O ( h 2 M ) , $$M=1,2,3,4$$ M = 1 , 2 , 3 , 4 , are provided.

Approximation of Solutions to Equations in Static Thermoelasticity

“…A one-dimensional integral representation with separated integrand of the action of Γ (k, ) on the tensor product generating functions (2.5) was obtained in [13]. We write…”

“…Replacing (3.8) and (3.9) in (3.1), we get approximation formulas for Γ (k, ) g suitable for fast computation. Numerical tests showing that these formulas are efficient and provide approximations of order O(h 2M ), M = 1, 2, 3, 4, are given in [13].…”

“…In [11] and [12], we applied this procedure to obtain cubature formulas for advection-diffusion operators over rectangular boxes in R n . Fast cubature formulas of elastic and hydrodynamic potentials were obtained in [13]. In [14] our approach was extended to parabolic problems.…”

Fast computation of the multidimensional fractional Laplacian

“…By combining cubature formulas for volume potentials based on approximate approximations with the strategy of separated representations (cf., e.g., [4], [12]), it is possible derive a method for approximating volume potentials which is accurate and fast also in the multidimensional case and provides approximation formulas of high order. This procedure was applied successfully for the fast integration of the harmonic [15], biharmonic [19], diffraction [18], elastic and hydrodynamic [20] potentials. In [16], [17] this approach was extended to parabolic problems.…”