scholarly journals Error estimates for the Ultra Weak Variational Formulation of the Helmholtz equation

2008 ◽  
Vol 42 (6) ◽  
pp. 925-940 ◽  
Author(s):  
Annalisa Buffa ◽  
Peter Monk
Keyword(s):  
Helmholtz Equation ◽  
Error Estimates ◽  
Variational Formulation

scholarly journals A Wavelet Method for the Cauchy Problem for the Helmholtz Equation

2012 ◽  
Vol 2012 ◽  
pp. 1-18
Author(s):  
Fang-Fang Dou ◽  
Chu-Li Fu
Keyword(s):  
Cauchy Problem ◽  
Exact Solution ◽  
Helmholtz Equation ◽  
Error Estimates ◽  
Fixed Frequency ◽  
Wavelet Method ◽  
Numerical Tests ◽  
The Cauchy Problem ◽  
Ill Posed

We consider a Cauchy problem for the Helmholtz equation at a fixed frequency. The problem is severely ill posed in the sense that the solution (if it exists) does not depend continuously on the data. We present a wavelet method to stabilize the problem. Some error estimates between the exact solution and its approximation are given, and numerical tests verify the efficiency and accuracy of the proposed method.

scholarly journals An Iterative Regularization Method to Solve the Cauchy Problem for the Helmholtz Equation

2014 ◽  
Vol 2014 ◽  
pp. 1-9
Author(s):  
Hao Cheng ◽  
Ping Zhu ◽  
Jie Gao
Keyword(s):  
Cauchy Problem ◽  
Exact Solution ◽  
Helmholtz Equation ◽  
Error Estimates ◽  
Regularization Method ◽  
A Priori ◽  
A Posteriori ◽  
Iterative Regularization ◽  
Regularization Parameters ◽  
The Cauchy Problem

A regularization method for solving the Cauchy problem of the Helmholtz equation is proposed. Thea priorianda posteriorirules for choosing regularization parameters with corresponding error estimates between the exact solution and its approximation are also given. The numerical example shows the effectiveness of this method.

OPTIMAL ERROR BOUND AND APPROXIMATION METHODS FOR A CAUCHY PROBLEM OF THE MODIFIED HELMHOLTZ EQUATION

2011 ◽  
Vol 09 (02) ◽  
pp. 305-315 ◽  
Author(s):  
AILIN QIAN ◽  
YUJIANG WU
Keyword(s):  
Cauchy Problem ◽  
Helmholtz Equation ◽  
Error Estimates ◽  
Error Bound ◽  
Optimality Theory ◽  
Tikhonov Regularization Method ◽  
Regularization Methods ◽  
Optimal Error ◽  
Modified Helmholtz Equation ◽  
Optimal Error Bound

We consider a Cauchy problem for a modified Helmholtz equation, especially when we give the optimal error bound for this problem. Some spectral regularization methods and a revised Tikhonov regularization method are used to stabilize the problem from the viewpoint of general regularization theory. Hölder-type stability error estimates are provided for these regularization methods. According to the optimality theory of regularization, the error estimates are order optimal.

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.

On regularization and error estimates for the Cauchy problem of the modified inhomogeneous Helmholtz equation

2016 ◽  
Vol 24 (5) ◽  
Author(s):  
Phan Trung Hieu ◽  
Pham Hoang Quan
Keyword(s):  
Cauchy Problem ◽  
Helmholtz Equation ◽  
Error Estimates ◽  
The Cauchy Problem

AbstractIn this paper, we consider the modified inhomogeneous Helmholtz equation Δ

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