8 research outputs found
Convergence of adaptive discontinuous galerkin methods
We develop a general convergence theory for adaptive discontinu-
ous Galerkin methods for elliptic PDEs covering the popular SIPG, NIPG and
LDG schemes as well as all practically relevant marking strategies. Another
key feature of the presented result is, that it holds for penalty parameters only
necessary for the standard analysis of the respective scheme. The analysis
is based on a quasi interpolation into a newly developed limit space of the
adaptively created non-conforming discrete spaces, which enables to generalise
the basic convergence result for conforming adaptive finite element methods by
Morin, Siebert, and Veeser [A basic convergence result for conforming adaptive
finite elements, Math. Models Methods Appl. Sci., 2008, 18(5), 707–737]
Convergence Analysis of the Lowest Order Weakly Penalized Adaptive Discontinuous Galerkin Methods
In this article, we prove convergence of the weakly penalized adaptive
discontinuous Galerkin methods. Unlike other works, we derive the contraction
property for various discontinuous Galerkin methods only assuming the
stabilizing parameters are large enough to stabilize the method. A central idea
in the analysis is to construct an auxiliary solution from the discontinuous
Galerkin solution by a simple post processing. Based on the auxiliary solution,
we define the adaptive algorithm which guides to the convergence of adaptive
discontinuous Galerkin methods
Convergent adaptive hybrid higher-order schemes for convex minimization
This paper proposes two convergent adaptive mesh-refining algorithms for the
hybrid high-order method in convex minimization problems with two-sided
p-growth. Examples include the p-Laplacian, an optimal design problem in
topology optimization, and the convexified double-well problem. The hybrid
high-order method utilizes a gradient reconstruction in the space of piecewise
Raviart-Thomas finite element functions without stabilization on triangulations
into simplices or in the space of piecewise polynomials with stabilization on
polytopal meshes. The main results imply the convergence of the energy and,
under further convexity properties, of the approximations of the primal resp.
dual variable. Numerical experiments illustrate an efficient approximation of
singular minimizers and improved convergence rates for higher polynomial
degrees. Computer simulations provide striking numerical evidence that an
adopted adaptive HHO algorithm can overcome the Lavrentiev gap phenomenon even
with empirical higher convergence rates
Convergence of adaptive discontinuous Galerkin and -interior penalty finite element methods for Hamilton--Jacobi--Bellman and Isaacs equations
We prove the convergence of adaptive discontinuous Galerkin and
-interior penalty methods for fully nonlinear second-order elliptic
Hamilton--Jacobi--Bellman and Isaacs equations with Cordes coefficients. We
consider a broad family of methods on adaptively refined conforming simplicial
meshes in two and three space dimensions, with fixed but arbitrary polynomial
degrees greater than or equal to two. A key ingredient of our approach is a
novel intrinsic characterization of the limit space that enables us to identify
the weak limits of bounded sequences of nonconforming finite element functions.
We provide a detailed theory for the limit space, and also some original
auxiliary functions spaces, that is of independent interest to adaptive
nonconforming methods for more general problems, including Poincar\'e and trace
inequalities, a proof of density of functions with nonvanishing jumps on only
finitely many faces of the limit skeleton, approximation results by finite
element functions and weak convergence results
Convergence of adaptive discontinuous Galerkin methods
We develop a general convergence theory for adaptive discontinuous Galerkin methods for elliptic PDEs covering the popular SIPG, NIPG and LDG schemes as well as all practically relevant marking strategies. Another key feature of the presented result is, that it holds for penalty parameters only necessary for the standard analysis of the respective scheme. The analysis is based on a quasi-interpolation into a newly developed limit space of the adaptively created non-conforming discrete spaces, which enables us to generalise the basic convergence result for conforming adaptive finite element methods by Morin, Siebert, and Veeser