An efficient meshfree point collocation moving least squares method to solve the interface problems with nonhomogeneous jump conditions

Abstract

We are going to study a simple and effective method for the numerical solution of the closed interface boundary value problem with both discontinuities in the solution and its derivatives. It uses a strong-form meshfree method based on the moving least squares (MLS) approximation. In this method, for the solution of elliptic equation, the second-order derivatives of the shape functions are needed in constructing the global stiffness matrix. It is well-known that the calculation of full derivatives of the MLS approximation, especially in high dimensions, is quite costly. In the current work, we apply the diffuse derivatives using an efficient technique. In this technique, we calculate the higher-order derivatives using the approximation of lower-order derivatives, instead of calculating directly derivatives. This technique can improve the accuracy of meshfree point collocation method for interface problems with nonhomogeneous jump conditions and can efficiently estimate diffuse derivatives of second-and higher-orders using only linear basis functions. To introduce the appropriate discontinuous shape functions in the vicinity of interface, we choose the visibility criterion method that modifies the support of weight function in MLS approximation and leads to an efficient computational procedure for the solution of closed interface problems. The proposed method is applied for elliptic and biharmonic interface problems. For the biharmonic equation, we use a mixed scheme, which replaces this equation by a coupled elliptic system. Also the application of the present method to elasticity equation with discontinuities in the coefficients across a closed interface has been provided. Representative numerical examples demonstrate the accuracy and robustness of the proposed methodology for the closed interface problems.

Optimal uniform error estimates for moving least‐squares collocation with application to option pricing under jump‐diffusion processes

“…This method has been applied successfully to approximate solutions of time‐independent [13, 14] and also time‐dependent PDEs [9]. The method is also applied to the numerical solution of mathematical models arising from engineering [15, 16] and also valuation of American options under the Black–Sholes model [17, 18].…”

“…16) which by compactness of the computational domain Ω and Ω ⊂ Ω leads to||u − Qu|| L ∞ ( Ω) ≤ ||u − Qu|| L ∞ (Ω) . (3.17)So, according to Theorem 2.3 and Equation (3.15) we arrive at…”

A unified algorithm for the selection of collocation stencils for convex, concave, and singular problems

“…For singular problems, the visibility criterion, introduced as part of the element free Galerkin (EFG) method, 14 has been proven effective for both meshless methods [14][15][16] and collocation methods. 2,[17][18][19] The concept of the visibility criterion is presented in Figure 1(A). Considering a domain Ω, only the support nodes X p which are "visible" from the collocation nodes X c are considered in the stencil.…”

“…For singular problems, the visibility criterion, introduced as part of the element free Galerkin (EFG) method, 14 has been proven effective for both meshless methods 14‐16 and collocation methods 2,17‐19 . The concept of the visibility criterion is presented in Figure 1(A).…”

Element‐free Galerkin method to the interface problems with application in electrostatic

“…In , Taleei and Dehghan combined the direct meshless local Petrov–Galerkin (MLPG) method with the visibility criterion technique for solving interface problems. Also, in , they used the visibility criterion method to modify the support of weight functions in moving least squares (MLS) approximation for solving interface problems with meshfree point collocation MLS method.…”