The perfectly matched layer for the ultra weak variational formulation of the 3D Helmholtz equation

2004 ◽  
Vol 61 (7) ◽  
pp. 1072-1092 ◽  
Author(s):  
Tomi Huttunen ◽  
Jari P. Kaipio ◽  
Peter Monk
Keyword(s):  
Helmholtz Equation ◽  
Variational Formulation ◽  
Perfectly Matched Layer

scholarly journals Dual variational methods for a nonlinear Helmholtz equation with sign-changing nonlinearity

2021 ◽  
Vol 60 (4) ◽  
Author(s):  
Rainer Mandel ◽  
Dominic Scheider ◽  
Tolga Yeşil
Keyword(s):  
Helmholtz Equation ◽  
Variational Methods ◽  
Variational Formulation ◽  
Morse Index ◽  
Existence Results ◽  
Sign Changes

AbstractWe prove new existence results for a nonlinear Helmholtz equation with sign-changing nonlinearity of the form $$\begin{aligned} - \Delta u - k^{2}u = Q(x)|u|^{p-2}u, \quad u \in W^{2,p}\left( {\mathbb {R}}^{N}\right) \end{aligned}$$ - Δ u - k 2 u = Q ( x ) | u | p - 2 u , u ∈ W 2 , p R N with $$k>0,$$ k > 0 , $$N \ge 3$$ N ≥ 3 , $$p \in \left[ \left. \frac{2(N+1)}{N-1},\frac{2N}{N-2}\right) \right. $$ p ∈ 2 ( N + 1 ) N - 1 , 2 N N - 2 and $$Q \in L^{\infty }({\mathbb {R}}^{N})$$ Q ∈ L ∞ ( R N ) . Due to the sign-changes of Q, our solutions have infinite Morse-Index in the corresponding dual variational formulation.

PML’s Optimal Choice in Solving Helmholtz Equation

2010 ◽  
Vol 39 ◽  
pp. 312-316
Author(s):  
Bo Zhang
Keyword(s):  
Boundary Conditions ◽  
Helmholtz Equation ◽  
Numerical Experiments ◽  
Accurate Result ◽  
Perfectly Matched Layer ◽  
Optimal Choice

To solve truncation questions of calculation area(unbounded) of Helmhotlz equation, Berenger first proposed concept of Perfectly Matched Layer(PML) in 1994, the method optimizes boundary conditions and reduces computation quantities greatly. By choosing constants p,d,e of PML parameters , we obtain an optimal PML parameter in this paper . The final numerical experiments show that the result obtained by the PML parameter is almost same as accurate result of references [4].

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