56 research outputs found
Enhanced RFB method
The residual-free bubble method (RFB) is a parameter-free stable finite element method that has been applied successfully to solve a wide range of boundary-value problems presenting multiple-scale behavior. If some local features of the solution are known a-priori, the RFB finite element space approximation properties can be increased by enriching it on some specific edges of the partition (see[7]). Based on such idea, we define and analyse the enhanced residual-free bubbles method for the solution of convection-dominated convection-diffusion problems in 2-D. Our a-priori analysis enlightens the limitations of the RFB method and the superior global convergence properties of the new method. The theoretical results are supported by extensive numerical experimentation.\ud
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The first author acknowledges the financial support of INdAM and EPSR
Conforming and nonconforming virtual element methods for elliptic problems
We present in a unified framework new conforming and nonconforming Virtual
Element Methods (VEM) for general second order elliptic problems in two and
three dimensions. The differential operator is split into its symmetric and
non-symmetric parts and conditions for stability and accuracy on their discrete
counterparts are established. These conditions are shown to lead to optimal
- and -error estimates, confirmed by numerical experiments on a set
of polygonal meshes. The accuracy of the numerical approximation provided by
the two methods is shown to be comparable
Discontinuous Galerkin Methods for Mass Transfer through Semi-Permeable Membranes
A discontinuous Galerkin (dG) method for the numerical solution of
initial/boundary value multi-compartment partial differential equation (PDE)
models, interconnected with interface conditions, is presented and analysed.
The study of interface problems is motivated by models of mass transfer of
solutes through semi-permeable membranes. More specifically, a model problem
consisting of a system of semilinear parabolic advection-diffusion-reaction
partial differential equations in each compartment, equipped with respective
initial and boundary conditions, is considered. Nonlinear interface conditions
modelling selective permeability, congestion and partial reflection are applied
to the compartment interfaces. An interior penalty dG method is presented for
this problem and it is analysed in the space-discrete setting. The a priori
analysis shows that the method yields optimal a priori bounds, provided the
exact solution is sufficiently smooth. Numerical experiments indicate agreement
with the theoretical bounds and highlight the stability of the numerical method
in the advection-dominated regime
A-posteriori error estimators and RFB
We derive a posteriori bounds for the residual-free bubble (RFB) method for the solution of convection-dominated diffusion equations. Both linear functional error control and energy norm error control are considered. The implementation of a reliable and efficient adaptive algorithm is discussed. Finally, we proposed an adaptive algorithm in which the local bubble stabilisation is automatically turned off ( derefinement) in large parts of the computational domain during the refinement process, without compromising the accuracy of the method.\ud
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The first author acknowledges the financial support of INdAM and EPSRC
Convergence of an adaptive discontinuous Galerkin method for elliptic interface problems
We prove a basic error contraction result of an adaptive discontinuous Galerkin method for an elliptic interface problem. The interface conditions considered model mass transfer of solutes through semi-permeable membranes and other filtering processes. The adaptive algorithm is based on a residual-type a posteriori error estimator, with a bulk refinement criterion. The a posteriori error bound is derived under the assumption that the triangulation is aligned with the interfaces although, crucially, extremely general curved element shapes are also allowed, resolving the interface geometry exactly. As a corollary, convergence of the adaptive discontinuous Galerkin method for non-essential Neumann-and/or Robin-type boundary conditions, posed on general curved boundaries, also follows. Numerical experiments are also presented. Keywords: Discontinuous Galerkin method, interface problem, a posteriori error bound, adaptivity, convergence analysis, a posteriori error analysis on curved domains
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