A Regularization Method for the Elliptic Equation with Inhomogeneous Source

We consider the following Cauchy problem for the elliptic equation with inhomogeneous source in a rectangular domain with Dirichlet boundary conditions at x = 0 and x = π. The problem is ill-posed. The main aim of this paper is to introduce a regularizationmethod and use it to solve the problem. Some sharp error estimates between the exact solution and its regularization approximation are given and a numerical example shows that the method works effectively.


Introduction
The Cauchy problem for the elliptic equation has been extensively investigated in many practical areas. For example, some problems relating to geophysics [1], plasma physics [2], and bioelectric field problems [3] are equivalent to solving the Cauchy problem for the elliptic equation. In this paper, we consider the following Cauchy problem for elliptic equation with nonhomogeneous source: where ∈ 2 (0, ), ∈ 2 (0, 1; 2 (0, )) are given. Problem (1)-(4) is well known to be ill-posed in the sense of Hadamard: a small perturbation in the data may cause dramatically large errors in the solution ( , ) for 0 < ≤ 1.
Very recently, in 2009, Hào and his group [23] applied the nonlocal-boundary value method to regularize the abstract homogeneous elliptic problem. This method is also given in [24] for solving an elliptic problem with homogeneous source in a cylindrical domain. A mollification regularization method for the Cauchy problem in a multidimensional case has been considered in the recent paper of Cheng and his group [25].
Although there are many papers on the homogeneous elliptic equation, the result on the inhomogeneous case is very scarce, while the inhomogeneous case is, of course, more general and nearer to practical application than the homogeneous one. Shortly, it allows the occurrence of some elliptic source which is inevitable in nature. The main aim of this paper is to present a simple and effective regularization method and investigate the error estimate between the regularization solution and the exact solution. In a sense, this paper may be an extension of many previous results.
The remainder of the paper is divided into two sections. In Section 2, we will study the regularization of problem (1)- (4) and obtain convergence estimates. In Section 3, a numerical test case for inhomogeneous problems is given to describe the effectiveness of our method.

Regularization and Error Estimate
By the method of separation of variables, the solution of problem (1)-(4) is given by We can see that the instability is caused by the fast growth of , > 0 as tends to infinity. Even though these exact Fourier coefficients , ( ) may tend to zero rapidly, in practice, performing classical calculation is impossible because the given data is usually diffused by a variety of reasons such as round-off error and measurement error. A small perturbation in the data can arbitrarily deduce a large error in the solution. Therefore, some special regularization methods are required. From (5), we replace the term that causes dramatically the increasing of the right side by several bounded approximations. We assume that the exact data ( ) and the measured data ( ) both belong to 2 (0, ) and satisfy ‖ − ‖ 2 ≤ where ‖ ⋅ ‖ 2 is the norm on 2 (0, ) and denotes the noise level, respectively. In the paper, we will use a modification method to regularize our problem. The regularized solution is given as follows: Here ∈ (0, 1) is a parameter regularization which depends on . The explicit error estimates including error estimates have been given according to some priori assumptions on the regularity of the exact solution.
Let V be the solution of problem (7) corresponding to the measured data . Then, it is given by We first have the following theorem.

Remark 3.
From (13), as → 1, the accuracy of regularized solution becomes progressively lower. To retain the continuous dependence of the solution at = 1, we introduce the following theorem.

Remark 6.
In this theorem, we require a condition on the Fourier expansion coefficient in (32). This condition is very difficult to check. To improve this, in the next theorem, we only require the assumption of , not to depend on the function .

A Numerical Experiment
To illustrate the theoretical results obtained before, we will discuss the corresponding numerical aspects in this section.