130 research outputs found
Convex Regularization Method for Solving Cauchy Problem of the Helmholtz Equation
In this paper, we introduce the Convex Regularization Method (CRM) for regularizing the (instability) solution of the Helmholtz equation with Cauchy data. The CRM makes it possible for the solution of Helmholtz equation to depend continuously on the small perturbations in the Cauchy data. In addition, the numerical computation of the reg- ularized Helmholtz equation with Cauchy data is stable, accurate and gives high rate of convergence of solution in Hilbert space. Undoubtedly, the error estimated analysis associated with CRM is minimal.Mathematics Subject Classi cation: 44B28; 44B30Keywords: Convex Regularization Method, ill-posed Helmholtz equation with Cauchy data, stable solutio
Approximation of mild solutions of the linear and nonlinear elliptic equations
In this paper, we investigate the Cauchy problem for both linear and
semi-linear elliptic equations. In general, the equations have the form
where is a positive-definite, self-adjoint operator with
compact inverse. As we know, these problems are well-known to be ill-posed. On
account of the orthonormal eigenbasis and the corresponding eigenvalues related
to the operator, the method of separation of variables is used to show the
solution in series representation. Thereby, we propose a modified method and
show error estimations in many accepted cases. For illustration, two numerical
examples, a modified Helmholtz equation and an elliptic sine-Gordon equation,
are constructed to demonstrate the feasibility and efficiency of the proposed
method.Comment: 29 pages, 16 figures, July 201
A new general filter regularization method for Cauchy problems for elliptic equations with a locally Lipschitz nonlinear source.
Up to now, studies on the semi-linear Cauchy problem for elliptic partial differential equations needed to assume that the source term present in the governing equation is a global Lipschitz function. The current paper is the first investigation to not only the more general but also the more practical case of interest when the source term is only a local Lipschitz function. In such a situation, the methods of solution from the previous studies with a global Lipschitz source term are not directly applicable and therefore, novel ideas and techniques need to be developed to tackle the local Lipschitz nonlinearity. This locally Lipschitz source arises in many applications of great physical interest governed by, for example, the sine-Gordon, Lane–Emden, Allen–Cahn and Liouville equations. The inverse problem is severely ill-posed in the sense of Hadamard by violating the continuous dependence upon the input Cauchy data. Therefore, in order to obtain a stable solution we consider theoretical aspects of regularization of the problem by a new generalized filter method. Under some priori assumptions on the exact solution, we prove and obtain rigorously convergence estimates
A finite element data assimilation method for the wave equation
We design a primal-dual stabilized finite element method for the numerical
approximation of a data assimilation problem subject to the acoustic wave
equation. For the forward problem, piecewise affine, continuous, finite element
functions are used for the approximation in space and backward differentiation
is used in time. Stabilizing terms are added on the discrete level. The design
of these terms is driven by numerical stability and the stability of the
continuous problem, with the objective of minimizing the computational error.
Error estimates are then derived that are optimal with respect to the
approximation properties of the numerical scheme and the stability properties
of the continuous problem. The effects of discretizing the (smooth) domain
boundary and other perturbations in data are included in the analysis.Comment: 23 page
Stabilised finite element methods for ill-posed problems with conditional stability
In this paper we discuss the adjoint stabilised finite element method
introduced in, E. Burman, Stabilized finite element methods for nonsymmetric,
noncoercive and ill-posed problems. Part I: elliptic equations, SIAM Journal on
Scientific Computing, and how it may be used for the computation of solutions
to problems for which the standard stability theory given by the Lax-Milgram
Lemma or the Babuska-Brezzi Theorem fails. We pay particular attention to
ill-posed problems that have some conditional stability property and prove
(conditional) error estimates in an abstract framework. As a model problem we
consider the elliptic Cauchy problem and provide a complete numerical analysis
for this case. Some numerical examples are given to illustrate the theory.Comment: Accepted in the proceedings from the EPSRC Durham Symposium Building
Bridges: Connections and Challenges in Modern Approaches to Numerical Partial
Differential Equation
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