Bhupen Deka

Optimal Error Estimates for Linear Parabolic Problems with Discontinuous Coefficients

A finite element discretization is proposed and analyzed for a linear parabolic problems with discontinuous coefficients. Due to low global regularity of the solution, it seems difficult to achieve optimal order of convergence with classical finite element methods [Numer. Math., 79 (1998), pp. 175--202]. In this paper, we have used a finite element discretization, where interface triangles are assumed to be curved triangles instead of straight triangles as in classical finite element methods. Optimal order error estimates in L2 and H1 norms are shown to hold even if the regularity of the solution is low on the whole domain. While the continuous time Galerkin method is discussed for the spatially discrete scheme, the discontinuous Galerkin method is analyzed for the fully discrete scheme. The interfaces and boundaries of the domains are assumed to be smooth for our purpose.

Finite element methods with numerical quadrature for elliptic problems with smooth interfaces

The purpose of this paper is to study the effect of the numerical quadrature on the finite element approximation to the exact solution of elliptic equations with discontinuous coefficients. Due to low global regularity of the solution, it seems difficult to achieve optimal order of convergence with classical finite element methods [Z. Chen, J. Zou, Finite element methods and their convergence for elliptic and parabolic interface problems, Numer. Math. 79 (1998) 175-202]. We derive error estimates in finite element method with quadrature for elliptic interface problems in a two-dimensional convex polygonal domain. Optimal order error estimates in L^2 and H^1 norms are shown to hold even if the regularity of the solution is low on the whole domain. Finally, numerical experiment for two dimensional test problem is presented in support of our theoretical findings.

L@ (L2) and L@ (H1) Norms Error Estimates in Finite Element Method for Linear Parabolic Interface Problems

We derive new a priori error estimates for linear parabolic equations with discontinuous coefficients. Due to low global regularity of the solutions the error analysis of the standard finite element method for parabolic problems is difficult to adopt for parabolic interface problems. A finite element procedure is, therefore, proposed and analyzed in this paper. We are able to show that the standard energy technique of finite element method for non-interface parabolic problems can be extended to parabolic interface problems if we allow interface triangles to be curved triangles. Optimal pointwise-in-time error estimates in the L 2(Ω) and H 1(Ω) norms are shown to hold for the semidiscrete scheme. A fully discrete scheme based on backward Euler method is analyzed and pointwise-in-time error estimates are derived. The interfaces are assumed to be arbitrary shape but smooth for our purpose.

Finite element methods for second order linear hyperbolic interface problems

Abstract The aim of this paper is to study finite element methods and their convergence for hyperbolic interface problems. Both semidiscrete and fully discrete schemes are analyzed. Optimal a priori error estimates in the L 2 and H 1 norms are derived for a finite element discretization where interface triangles are assumed to be curved triangles instead of straight triangles. The interfaces and boundaries of the domains are assumed to be smooth for our purpose.