Consistent weak forms for meshfree methods: Full realization of h-refinement, p-refinement, and a-refinement in strong-type essential boundary condition enforcement

2021 ◽  
Vol 373 ◽  
pp. 113448
Author(s):  
Michael Hillman ◽  
Kuan-Chung Lin
Keyword(s):  
Boundary Condition ◽  
Meshfree Methods ◽  
Strong Type ◽  
Essential Boundary Condition

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.

New Method for Implementing Essential Boundary Condition to Element-Free Galerkin Method

2010 ◽  
Author(s):  
Ayumu Saitoh ◽  
Taku Itoh ◽  
Nobuyuki Matsui ◽  
Atsushi Kamitani ◽  
Theodore E. Simos ◽  
...  
Keyword(s):  
Boundary Condition ◽  
Galerkin Method ◽  
New Method ◽  
Element Free Galerkin Method ◽  
Essential Boundary Condition ◽  
Element Free Galerkin ◽  
Element Free
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