Some Error Estimates for the Discretization of Parabolic Equations on General Multidimensional Nonconforming Spatial Meshes

摘要

This work is devoted to error estimates for the discretization of parabolic equations on general nonconforming spatial meshes in several space dimensions. These meshes have been recently used to approximate stationary anisotropic heterogeneous diffusion equations and nonlinear equations. We present an implicit time discretization scheme based on an orthogonal projection of the exact initial value. We prove that, when the discrete flux is calculated using a stabilized discrete gradient, the convergence order is h-p + k, where h-o (resp. k) is the mesh size of the spatial (resp. time) discretization. This estimate is valid for discrete norms L~∞(0,7'; H_0~2(Ω)) and W~(1,∞)(0, T; L~2(Ω)) under the regularity assumption u ∈ C~2([0,T];C~2(Ω)) for the exact solution u. These error estimates are useful because they allow to obtain approximations to the exact solution and its first derivatives of order h_D + k.