Abstract
The elliptic equations with discontinuous coefficients are often used to describe the problems of the multiple materials or fluids with different densities or conductivities or diffusivities. In this paper we develop a partially penalty immersed finite element (PIFE) method on triangular grids for anisotropic flow models, in which the diffusion coefficient is a piecewise definite-positive matrix. The standard linear Crouzeix-Raviart type finite element space is used on non-interface elements and the piecewise linear Crouzeix-Raviart type immersed finite element (IFE) space is constructed on interface elements. The piecewise linear functions satisfying the interface jump conditions are uniquely determined by the integral averages on the edges as degrees of freedom. The PIFE scheme is given based on the symmetric, nonsymmetric or incomplete interior penalty discontinuous Galerkin formulation. The solvability of the method is proved and the optimal error estimates in the energy norm are obtained. Numerical experiments are presented to confirm our theoretical analysis and show that the newly developed PIFE method has optimal-order convergence in the L^{2} norm as well. In addition, numerical examples also indicate that this method is valid for both the isotropic and the anisotropic elliptic interface problems.
Highlights
The elliptic equations with discontinuous coefficients are often used to describe phenomena appearing in material sciences and fluid dynamics when there are two or more distinct materials or fluids with different densities or conductivities or diffusivities
For anisotropic elliptic interface problems, we presented the partially penalty immersed finite element (IFE) method in [ ] by adding two penalty terms on the common edges of the adjacent interface elements to restrict the function jumps, and deriving the optimal-order error estimates
We develop the partially penalty immersed finite element (PIFE) method with the Crouzeix-Raviart type polynomial spaces to solve the anisotropic flow models in which the diffusion coefficient is a piecewise definite-positive matrix, we consider isotropic elliptic problems with piecewise scalar coefficients in major length for simplicity
Summary
The elliptic equations with discontinuous coefficients are often used to describe phenomena appearing in material sciences and fluid dynamics when there are two or more distinct materials or fluids with different densities or conductivities or diffusivities. There is a small region in these two types of interface elements, Tr = T \ + ∩ T + \ – ∩ T – whose area is of order O(h ) since the interface is a C curve and the interface ∩ T is perturbed in a magnitude of O(h ) [ ] Such a perturbation will only affect the solution and the interpolation function to an order of O(h ), which will not impact on the convergence accuracy of the method whose approximation spaces are selected as piecewise linear polynomials in this paper. On each of these interface triangular elements, T ∈ Thi, for given values Vi, i = , , , the piecewise linear function φ can be defined by φ.
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