Numerical Method for Solving Matrix Coefficient Elliptic Equation on Irregular Domains with Sharp-Edged Boundaries

We present a new second-order accurate numerical method for solving matrix coefficient elliptic equation on irregular domains with sharp-edged boundaries. Nontraditional finite element method with non-body-fitting grids is implemented on a fictitious domain in which the irregular domains are embedded. First we set the function and coefficient in the fictitious part, and the nonsmooth boundary is then treated as an interface. The emphasis is on the construction of jump conditions on the interface; a special position for the ghost point is chosen so that the method is more accurate. The test function basis is chosen to be the standard finite element basis independent of the interface, and the solution basis is chosen to be piecewise linear satisfying the jump conditions across the interface. This is an efficient method for dealing with elliptic equations in irregular domains with non-smooth boundaries, and it is able to treat the general case of matrix coefficient. The complexity and computational expense in mesh generation is highly decreased, especially for moving boundaries, while robustness, efficiency, and accuracy are promised. Extensive numerical experiments indicate that this method is second-order accurate in the norm in both two and three dimensions and numerically very stable. 1. Introduction Let ( ) be an open-bounded domain with a Lipschitz continuous boundary . We consider the variable coefficient elliptic equation where refers to the spatial variable, is the gradient operator, and the right-hand side is assumed to lie in . The coefficient is a matrix that is uniformly elliptic, and its entries are continuously differentiable on . For a given function on the boundary , the Dirichlet boundary condition is prescribed as Elliptic partial differential equations are often used to construct models of the most basic theories underlying physics and engineering, such as electromagnetism, material science, and fluid dynamics. Different kinds of boundary conditions arise with the wide range of applications, such as Dirichlet boundary condition, Neumann boundary condition, and Robin boundary condition. Elliptic equation on irregular domains has been studied by many researchers and several techniques have been developed. Finite element methods use a mesh triangulation to capture the boundary [1–4]. However, in many situations, such as when the boundary is moving, the mesh generation may be both computational expensive and challenging. A more preferred method is to combine the Cartesian grid method with level-set approach [5–7] to capture the boundary.