Anisotropic mesh refinement in polyhedral domains: error estimates with data in L^2(Ω)
Provides rigorous error bounds for anisotropic meshes in 3D polyhedral domains, enabling further analysis in optimal control and edge element methods.
The paper proves error estimates for finite element solutions of the Poisson equation on anisotropic graded meshes in 3D polyhedral domains, using a new quasi-interpolation operator to handle L^2 data. The results enable extensions to optimal control problems and simplify proofs of discrete compactness for edge elements.
The paper is concerned with the finite element solution of the Poisson equation with homogeneous Dirichlet boundary condition in a three-dimensional domain. Anisotropic, graded meshes from a former paper are reused for dealing with the singular behaviour of the solution in the vicinity of the non-smooth parts of the boundary. The discretization error is analyzed for the piecewise linear approximation in the H^1(Ω)- and L^2(Ω)-norms by using a new quasi-interpolation operator. This new interpolant is introduced in order to prove the estimates for L^2(Ω)-data in the differential equation which is not possible for the standard nodal interpolant. These new estimates allow for the extension of certain error estimates for optimal control problems with elliptic partial differential equation and for a simpler proof of the discrete compactness property for edge elements of any order on this kind of finite element meshes.