1,652 research outputs found
How a nonconvergent recovered Hessian works in mesh adaptation
Hessian recovery has been commonly used in mesh adaptation for obtaining the
required magnitude and direction information of the solution error.
Unfortunately, a recovered Hessian from a linear finite element approximation
is nonconvergent in general as the mesh is refined. It has been observed
numerically that adaptive meshes based on such a nonconvergent recovered
Hessian can nevertheless lead to an optimal error in the finite element
approximation. This also explains why Hessian recovery is still widely used
despite its nonconvergence. In this paper we develop an error bound for the
linear finite element solution of a general boundary value problem under a mild
assumption on the closeness of the recovered Hessian to the exact one.
Numerical results show that this closeness assumption is satisfied by the
recovered Hessian obtained with commonly used Hessian recovery methods.
Moreover, it is shown that the finite element error changes gradually with the
closeness of the recovered Hessian. This provides an explanation on how a
nonconvergent recovered Hessian works in mesh adaptation.Comment: Revised (improved proofs and a better example
Convergence of adaptive mixed finite element method for convection-diffusion-reaction equations
We prove the convergence of an adaptive mixed finite element method (AMFEM)
for (nonsymmetric) convection-diffusion-reaction equations. The convergence
result holds from the cases where convection or reaction is not present to
convection-or reaction-dominated problems. A novel technique of analysis is
developed without any quasi orthogonality for stress and displacement
variables, and without marking the oscillation dependent on discrete solutions
and data. We show that AMFEM is a contraction of the error of the stress and
displacement variables plus some quantity. Numerical experiments confirm the
theoretical results.Comment: arXiv admin note: text overlap with arXiv:1312.645
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