152 research outputs found
Adaptive vertex-centered finite volume methods for general second-order linear elliptic PDEs
We prove optimal convergence rates for the discretization of a general
second-order linear elliptic PDE with an adaptive vertex-centered finite volume
scheme. While our prior work Erath and Praetorius [SIAM J. Numer. Anal., 54
(2016), pp. 2228--2255] was restricted to symmetric problems, the present
analysis also covers non-symmetric problems and hence the important case of
present convection
On the convergence of adaptive nonconforming finite element methods for a class of convex variational problems
We formulate and analyze an adaptive nonconforming finite element method for the solution of convex variational problems. The class of minimization problems we admit includes highly singular problems for which no Euler–Lagrange equation (or inequality) is available. As a consequence, our arguments only use the structure of the energy functional. We are nevertheless able to prove convergence of an adaptive algorithm, using even refinement indicators that are not reliable error indicators
Reliable and efficient a posteriori error estimation for adaptive IGA boundary element methods for weakly-singular integral equations
We consider the Galerkin boundary element method (BEM) for weakly-singular
integral equations of the first-kind in 2D. We analyze some residual-type a
posteriori error estimator which provides a lower as well as an upper bound for
the unknown Galerkin BEM error. The required assumptions are weak and allow for
piecewise smooth parametrizations of the boundary, local mesh-refinement, and
related standard piecewise polynomials as well as NURBS. In particular, our
analysis gives a first contribution to adaptive BEM in the frame of
isogeometric analysis (IGABEM), for which we formulate an adaptive algorithm
which steers the local mesh-refinement and the multiplicity of the knots.
Numerical experiments underline the theoretical findings and show that the
proposed adaptive strategy leads to optimal convergence
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