238 research outputs found
Virtual Enriching Operators
We construct bounded linear operators that map conforming Lagrange
finite element spaces to conforming virtual element spaces in two and
three dimensions. These operators are useful for the analysis of nonstandard
finite element methods
Adaptive meshless centres and RBF stencils for Poisson equation
We consider adaptive meshless discretisation of the Dirichlet problem for Poisson equation based on numerical differentiation stencils obtained with the help of radial basis functions. New meshless stencil selection and adaptive refinement algorithms are proposed in 2D. Numerical experiments show that the accuracy of the solution is comparable with, and often better than that achieved by the mesh-based adaptive finite element method
A locally modified second-order finite element method for interface problems
The locally modified finite element method, which is introduced in [Frei,
Richter: SINUM 52(2014), p. 2315-2334] is a simple fitted finite element method
that is able to resolve weak discontinuities in interface problems. The method
is based on a fixed structured coarse mesh, which is then refined into
sub-elements to resolve an interior interface. In this work, we extend the
locally modified finite element method to second order using an isoparametric
approach in the interface elements. Thereby we need to take care that the
resulting curved edges do not lead to degenerate sub-elements. We prove optimal
a priori error estimates in the -norm and in a modified energy norm, as
well as a reduced convergence order of in the standard
-norm. Finally, we present numerical examples to substantiate the
theoretical findings
Computation and verification of Lyapunov functions
Lyapunov functions are an important tool to determine the basin of attraction of equilibria in Dynamical Systems through their sublevel sets. Recently, several numerical construction methods for Lyapunov functions have been proposed, among them the RBF (Radial Basis Function) and CPA (Continuous Piecewise Affine) methods. While the first method lacks a verification that the constructed function is a valid Lyapunov function, the second method is rigorous, but computationally much more demanding. In this paper, we propose a combination of these two methods, using their respective strengths: we use the RBF method to compute a potential Lyapunov function. Then we interpolate this function by a CPA function. Checking a finite number of inequalities, we are able to verify that this interpolation is a Lyapunov function. Moreover, sublevel sets are arbitrarily close to the basin of attraction. We show that this combined method always succeeds in computing and verifying a Lyapunov function, as well as in determining arbitrary compact subsets of the basin of attraction. The method is applied to two examples
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