8,932 research outputs found
A posteriori error estimates for the virtual element method
An a posteriori error analysis for the virtual element method (VEM) applied to general elliptic problems is presented. The resulting error estimator is of residual-type and applies on very general polygonal/polyhedral meshes. The estimator is fully computable as it relies only on quantities available from the VEM solution, namely its degrees of freedom and element-wise polynomial projection. Upper and lower bounds of the error estimator with respect to the VEM approximation error are proven. The error estimator is used to drive adaptive mesh refinement in a number of test problems. Mesh adaptation is particularly simple to implement since elements with consecutive co-planar edges/faces are allowed and, therefore, locally adapted meshes do not require any local mesh post-processing
A priori and a posteriori error analysis for a VEM discretization of the convection-diffusion eigenvalue problem
In this paper we propose and analyze a virtual element method for the two
dimensional non-symmetric diffusion-convection eigenvalue problem in order to
derive a priori and a posteriori error estimates. Under the classic assumptions
of the meshes, and with the aid of the classic theory of compact operators, we
prove error estimates for the eigenvalues and eigenfunctions. Also, we develop
an a posteriori error estimator which, in one hand, results to be reliable and
on the other, with standard bubble functions arguments, also results to be
efficient. We test our method on domains where the complex eigenfunctions are
not sufficiently regular, in order to assess the performance of the estimator
that we compare with the uniform refinement given by the a priori analysi
Adaptive non-hierarchical Galerkin methods for parabolic problems with application to moving mesh and virtual element methods
We present a posteriori error estimates for inconsistent and non-hierarchical Galerkin methods for linear parabolic problems, allowing them to be used in conjunction with very general mesh modification for the first time. We treat schemes which are non-hierarchical in the sense that the spatial Galerkin spaces between time-steps may be completely unrelated from one another. The practical interest of this setting is demonstrated by applying our results to finite element methods on moving meshes and using the estimators to drive an adaptive algorithm based on a virtual element method on a mesh of arbitrary polygons. The a posteriori error estimates, for the error measured in the L2(H1) and L∞(L2) norms, are derived using the elliptic reconstruction technique in an abstract framework designed to precisely encapsulate our notion of inconsistency and non-hierarchicality and requiring no particular compatibility between the computational meshes used on consecutive time-steps, thereby significantly relaxing this basic assumption underlying previous estimates
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