|Name of the article||On the integral equation approach for solution of a Neumann boundary value problem for an elliptic equation with variable coefficients|
|Abstract||There are a lot of different physical processes modeling of which lead us to a differential problems for elliptic equations with variable coefficients. An example of such process is electrical impedance tomography (EIT) – a medical imaging technique in which an image of the conductivity of some part of the body is determined from electrical surface measurements. Applications of EIT can be found in medicine (lung function monitoring, skin cancer detection), nondestructive testing (determination of cracks in materials) or geophysics. Elliptic equations with variable coefficients often appear in problems for heterogeneous environments (for instance, functionally graded materials).
It is well known that a boundary value problem for a differential partial equation (e.g. the Neumann boundary value problem for Laplace equation) can be reduced to a boundary integral equation and then numerically solved. However, this approach requires availability of a fundamental solution for main differential equation. Unfortunately, in the general case, the fundamental solution for elliptic equations with variable coefficients is unknown that makes it impossible to consider an equivalent boundary integral equation. One of options for solving this problem is to use a parametrix (Levi function) that transforms the differential problem to the boundary-integral equation (BDIE).
Two types of BDIEs can be distinguished. BDIE is called united if the unknown function u on the boundary is just a trace of the function u in the domain as opposed to the segregated BDIE, where the unknown boundary functions are considered as formally independent of the unknown variables in the domain. The integral equations of the first type can be obtained as a consequence of Green’s formula application to the differential operator of the equation taking into account the boundary condition. Using the parametrix-based potential operators we can obtain a system of BDIEs of the second type. The last approach is used in this paper.
We consider the Neumann boundary value problem for an elliptic equation with variable coefficients in a bounded simply connected domain. It is known that the solution u of the problem can determined uniquely up to an additive constant. A uniqueness of the unknown function u can be reached by applying an additional condition. We assume that the solution domain contains an origin of coordinates and the solution function equals zero at that point. Since the fundamental solution for is in general not explicitly known, as it was mentioned above, we use the parametrix.
Using an indirect integral equation approach the solution of the Neumann problem is represented as a sum of the volume and the single layer potentials (based on the parametrix) with unknown densities over domain and boundary. Substituting the representation of the solution to the differential equation and Neumann boundary condition, taking into account Levi function properties and jump relation property of the normal derivative of the single layer potential, we obtain a system of BDIEs of the second kind. Further we consider the modified system (to satisfy the uniqueness condition), assume that our simply connected domain is symmetric relative to origin and the boundary curve has the parametric representation. Making the change of variables in the double integrals the system of integral equations can be rewritten in the parameterized form.
The strong singularity in one of the kernels can be split using a statement in which a vector function is represented via its normal and tangential vectors. Employing the quadratures for continuous and strong singular integrands together with collocating the approximating equations at the quadrature points lead to the fully discretized system. Having the densities values we can find an approximate solution in the domain that is confirmed by numerical experiments.
|PDF format||Beshley A. |