6 research outputs found
Polynomial Particular Solutions for Solving Elliptic Partial Differential Equations
In the past, polynomial particular solutions have been obtained for certain types of partial differential operators without convection terms. In this paper, a closed-form particular solution for more general partial differential operators with constant coefficients has been derived for polynomial basis functions. The newly derived particular solution is further coupled with the method of particular solutions (MPS) for numerically solving a large class of elliptic partial differential equations. In contrast to the use of Chebyshev polynomial basis functions, the proposed approach is more flexible in selecting the collocation points inside the domain. The polynomial basis functions are well-known for yielding ill-conditioned systems when their order becomes large. The multiple scale technique is applied to circumvent the difficulty of ill-conditioning problem. Five numerical examples are presented to demonstrate the effectiveness of the proposed algorithm
The Method of Particular Solutions Using Trigonometric Basis Functions
In this paper, the method of particular solutions (MPS) using trigonometric functions as the basis functions is proposed to solve two-dimensional elliptic partial differential equations. The inhomogeneous term of the governing equation is approximated by Fourier series and the closed-form particular solutions of trigonometric functions are derived using the method of undetermined coefficients. Once the particular solutions for the trigonometric basis functions are derived, the standard MPS can be applied for solving partial differential equations. In comparing with the use of radial basis functions and polynomials in the MPS, our proposed approach provides another simple approach to effectively solving two-dimensional elliptic partial differential equations. Five numerical examples are provided in this paper to validate the merits of the proposed meshless method
Fast, high-order numerical evaluation of volume potentials via polynomial density interpolation
This article presents a high-order accurate numerical method for the
evaluation of singular volume integral operators, with attention focused on
operators associated with the Poisson and Helmholtz equations in two
dimensions. Following the ideas of the density interpolation method for
boundary integral operators, the proposed methodology leverages Green's third
identity and a local polynomial interpolant of the density function to recast
the volume potential as a sum of single- and double-layer potentials and a
volume integral with a regularized (bounded or smoother) integrand. The layer
potentials can be accurately and efficiently evaluated everywhere in the plane
by means of existing methods (e.g.\ the density interpolation method), while
the regularized volume integral can be accurately evaluated by applying
elementary quadrature rules. We describe the method both for domains meshed by
mapped quadrilaterals and triangles, introducing for each case (i)
well-conditioned methods for the production of certain requisite source
polynomial interpolants and (ii) efficient translation formulae for polynomial
particular solutions. Compared to straightforwardly computing corrections for
every singular and nearly-singular volume target, the method significantly
reduces the amount of required specialized quadrature by pushing all singular
and near-singular corrections to near-singular layer-potential evaluations at
target points in a small neighborhood of the domain boundary. Error estimates
for the regularization and quadrature approximations are provided. The method
is compatible with well-established fast algorithms, being both efficient not
only in the online phase but also to set-up. Numerical examples demonstrate the
high-order accuracy and efficiency of the proposed methodology
Fast, high-order numerical evaluation of volume potentials via polynomial density interpolation
This article presents a high-order accurate numerical method for the evaluation of singular volume integral operators, with attention focused on operators associated with the Poisson and Helmholtz equations in two dimensions. Following the ideas of the density interpolation method for boundary integral operators, the proposed methodology leverages Green's third identity and a local polynomial interpolant of the density function to recast the volume potential as a sum of single- and double-layer potentials and a volume integral with a regularized (bounded or smoother) integrand. The layer potentials can be accurately and efficiently evaluated everywhere in the plane by means of existing methods (e.g. the density interpolation method), while the regularized volume integral can be accurately evaluated by applying elementary quadrature rules. Compared to straightforwardly computing corrections for every singular and nearly-singular volume target, the method significantly reduces the amount of required specialized quadrature by pushing all singular and near-singular corrections to near-singular layer-potential evaluations at target points in a small neighborhood of the domain boundary. Error estimates for the regularization and quadrature approximations are provided. The method is compatible with well-established fast algorithms, being both efficient not only in the online phase but also to set-up. Numerical examples demonstrate the high-order accuracy and efficiency of the proposed methodology; applications to inhomogeneous scattering are presented.</p