A Discontinuous Finite Element Method For Anisotropic Elliptic Interface Problem

The anisotropic interface problems can described by the following equation withdiscontinuous coefcients: satisfyThe intermittent domain of discontinuous coefcients Γ called interface, andthe second-order tensor matrix B characterize anisotropic media. On the interface,the solution must satisfy the interface jump conditions(conservation law). If the in-terface is smooth enough, then the solution of the interface problem is also smoothin individual regions where the coefcient is smooth, but due to the jump of thecoefcient across the interface, the global regularity is usually low, and the solutionusually belongs to H1+α(),0≤α <1. The interface problems play a crucial rolewhen it accurately express the actual physical process, such as in material sciencesand fuid dynamics when two or more distinct materials or fuids with diferentdensities or difusivity are involved. Therefore, it is essential to establish efectivenumerical simulation and strict theoretical system for such problems.However, due to the low global regularity and the irregular geometry, it is dif-fcult to establish an appropriate system for numerical simulation and numerical analysis.Based on previous work, by constructing immersed interface fnite elementspace and taking advantage of discontinuous fnite element combines ideas, we pro-pose a partial discontinuous fnite element method for the anisotropic second orderelliptic interface problems. The main contents are as follows:1.We use uniform rectangular grid on domain.For interface elements, accord-ing to jump conditions and vertex values, we construct piecewise bilinear functionspace and prove it is unique solvability. For non-interface elements, we apply well-known bilinear interpolation space.2.To weaken the strongly non-coordination on the edge of interface elements,we add penalty terms. Combining with discontinuous fnite element thoughts, weconstruct the discontinuous fnite element formulation, and prove the existence anduniqueness of formulation at the same time.3.By using Strong Lemma, scale demonstration, trace theorem, Young’s in-equality, dual technology demonstration and other numerical analysis, we prove thatthe solution of immersed interface fnite element has the optimal approximation ac-curacy in H1andL2, and the order of the convergence is O(h) and O(h2)respectively.It is a good solution to solve the numerical simulation under rectangular for anisotropicfow model in porous media.