An analysis for the convergence order of gradient schemes for semilinear parabolic equations

Gradient schemes are numerical methods, which can be conforming and nonconforming, have been recently developed in Droniou et al. (2013), Droniou et al. (2015), Eymard et al. (2012) and references therein to approximate different types of partial differential equations. They are written in a discrete variational formulation and based on the approximation of functions and gradients. The aim of the present paper is to provide gradient schemes along with an analysis for the convergence order of these schemes for semilinear parabolic equations in any space dimension. We present three gradient schemes. The first two schemes are nonlinear whereas the third one is linear. The existence and uniqueness of the discrete solutions for the first two schemes is proved, thanks to the use of the method of contractive mapping, under the assumption that the mesh size of the time discretization k is small, whereas the existence and uniqueness of the discrete solution for the third scheme is proved for arbitrary k. We provide a convergence rate analysis in discrete semi-norms of L∞(H1) and W1,2(L2) and in the norm of L∞(L2). We prove that the order in space is the same one proved in Eymard et al. (2012) when approximating elliptic equations and one or two in time. The existence, uniqueness, and the convergence results stated above do not require any relation between spacial and temporal discretizations. As an application of these results, we focus on the gradient schemes which use the discrete gradient introduced recently in the SUSHI method (Eymard et al., 2010) and we provide some numerical tests.