Essential boundary condition enforcement in meshless methods: boundary flux collocation method

2001 ◽  
Vol 53 (3) ◽  
pp. 499-514 ◽  
Author(s):  
Cheng-Kong C. Wu ◽  
Michael E. Plesha
Keyword(s):  
Boundary Condition ◽  
Collocation Method ◽  
Meshless Methods ◽  
Essential Boundary Condition ◽  
Boundary Flux

An Efficient Radial Basis Collocation Method for the Boundary Condition Identification of the Inverse Wave Problem

2018 ◽  
Vol 10 (01) ◽  
pp. 1850010 ◽  
Author(s):  
Lihua Wang ◽  
Zhihao Qian ◽  
Zhen Wang ◽  
Yukui Gao ◽  
Yongbo Peng
Keyword(s):  
Boundary Condition ◽  
Collocation Method ◽  
Exponential Convergence ◽  
Initial Conditions ◽  
Identification Problem ◽  
Iteration Scheme ◽  
Wave Problem ◽  
Noisy Input ◽  
Radial Basis ◽  
Convergence Study

A meshfree collocation method using radial basis functions is proposed to identify the information of the inaccessible part of the boundary in the inverse wave problem. Tikhonov regularization associated with L-curve criterion is introduced to handle the ill-conditioned resulting matrix originated from the noisy input data. The method of Lagrange multiplier is employed to derive the appropriate weights for the known boundary condition and additional measured condition to achieve the optimum convergence. Iteration scheme can be avoided in the collocation algorithm which ensures efficiency, and the convergence study shows the exponential convergence for the solutions of this inverse wave problem. Numerical simulations demonstrate that stable and highly accurate results can be obtained in the boundary condition identification problem even when high noise is involved in the additional condition, known boundary condition or initial conditions.

scholarly journals Penalty method with Crouzeix–Raviart approximation for the Stokes equations under slip boundary condition

2019 ◽  
Vol 53 (3) ◽  
pp. 869-891
Author(s):  
Takahito Kashiwabara ◽  
Issei Oikawa ◽  
Guanyu Zhou
Keyword(s):  
Boundary Condition ◽  
Finite Element ◽  
Stokes Equations ◽  
Slip Boundary Condition ◽  
Trace Operator ◽  
Essential Boundary Condition ◽  
Slip Boundary ◽  
Polyhedral Domain ◽  
Discrete Counterpart ◽  
Original Domain

The Stokes equations subject to non-homogeneous slip boundary conditions are considered in a smooth domain Ω ⊂ ℝN (N = 2,3). We propose a finite element scheme based on the nonconforming P1/P0 approximation (Crouzeix–Raviart approximation) combined with a penalty formulation and with reduced-order numerical integration in order to address the essential boundary condition u · n∂Ω = g on ∂Ω. Because the original domain Ω must be approximated by a polygonal (or polyhedral) domain Ωh before applying the finite element method, we need to take into account the errors owing to the discrepancy Ω ≠ Ωh, that is, the issues of domain perturbation. In particular, the approximation of n∂Ω by n∂Ωh makes it non-trivial whether we have a discrete counterpart of a lifting theorem, i.e., continuous right inverse of the normal trace operator H1 (Ω)N → H1/2(∂Ω); u ↦ u⋅n∂Ω. In this paper we indeed prove such a discrete lifting theorem, taking advantage of the nonconforming approximation, and consequently we establish the error estimates O(hα + ε) and O(h2α + ε) for the velocity in the H1- and L2-norms respectively, where α = 1 if N = 2 and α = 1/2 if N = 3. This improves the previous result [T. Kashiwabara et al., Numer. Math. 134 (2016) 705–740] obtained for the conforming approximation in the sense that there appears no reciprocal of the penalty parameter ϵ in the estimates.

Development of the Smoothed Natural Neighbor Petrov-Galerkin Method

2012 ◽  
Vol 201-202 ◽  
pp. 198-201
Author(s):  
Kai Wang ◽  
Shen Jie Zhou ◽  
Zhi Feng Nie
Keyword(s):  
Boundary Condition ◽  
Galerkin Method ◽  
Accurate Result ◽  
Computational Cost ◽  
Smoothing Technique ◽  
Essential Boundary Condition ◽  
Natural Neighbor ◽  
Strain Smoothing ◽  
Mlpg Method ◽  
Domain Integration

The strain smoothing technique is employed in the natural neighbor Petrov-Galerkin method (NNPG), and the so-called smoothed natural neighbor Petrov-Galerkin method is proposed and studied. This method inherits the advantages of the generalized MLPG method and possesses the easy imposition of essential boundary condition and the domain integration is completely avoided. In comparison with the traditional NNPG, the smoothed natural neighbor Petrov-Galerkin method can obtained more stable and accurate result without increasing the computational cost.

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