Abstract
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 L∞ norm in both two and three dimensions and numerically very stable.
Highlights
IntroductionThe nontraditional finite element method is originally developed in [26,27,28,29,30,31,32] for solving elliptic or elasticity equations with sharp-edged interfaces
Let Ω− ⊂ Rd (d = 2, 3) be an open-bounded domain with a Lipschitz continuous boundary Γ.We consider the variable coefficient elliptic equation−∇ ⋅ (β− (x) ∇u− (x)) = f (x) in Ω−, (1)where x = (x1, . . . , xd) refers to the spatial variable, ∇ is the gradient operator, and the right-hand side f(x) is assumed to lie in L2(Ω−)
We present a new second-order accurate numerical method for solving matrix coefficient elliptic equation on irregular domains with sharp-edged boundaries
Summary
The nontraditional finite element method is originally developed in [26,27,28,29,30,31,32] for solving elliptic or elasticity equations with sharp-edged interfaces This method uses non-body-fitting Cartesian grids and uses different basis for the solution and test function; its linear system is independent of jump condition. In this way, it is easy to implement, especially for moving interface. By implementing the nontraditional finite element method on the fictitious domain containing the irregular domains which are under research, we overcome the difficulty of capturing the sharp-edged boundary with high accuracy Both two- and three-dimensional models are studied.
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