journal
https://read.qxmd.com/read/29375159/discrete-maximal-regularity-of-time-stepping-schemes-for-fractional-evolution-equations
#21
JOURNAL ARTICLE
Bangti Jin, Buyang Li, Zhi Zhou
In this work, we establish the maximal [Formula: see text]-regularity for several time stepping schemes for a fractional evolution model, which involves a fractional derivative of order [Formula: see text], [Formula: see text], in time. These schemes include convolution quadratures generated by backward Euler method and second-order backward difference formula, the L1 scheme, explicit Euler method and a fractional variant of the Crank-Nicolson method. The main tools for the analysis include operator-valued Fourier multiplier theorem due to Weis (Math Ann 319:735-758, 2001...
2018: Numerische Mathematik
https://read.qxmd.com/read/29151623/fast-interior-point-solution-of-quadratic-programming-problems-arising-from-pde-constrained-optimization
#22
JOURNAL ARTICLE
John W Pearson, Jacek Gondzio
Interior point methods provide an attractive class of approaches for solving linear, quadratic and nonlinear programming problems, due to their excellent efficiency and wide applicability. In this paper, we consider PDE-constrained optimization problems with bound constraints on the state and control variables, and their representation on the discrete level as quadratic programming problems. To tackle complex problems and achieve high accuracy in the solution, one is required to solve matrix systems of huge scale resulting from Newton iteration, and hence fast and robust methods for these systems are required...
2017: Numerische Mathematik
https://read.qxmd.com/read/29151622/a-posteriori-error-estimates-for-the-virtual-element-method
#23
JOURNAL ARTICLE
Andrea Cangiani, Emmanuil H Georgoulis, Tristan Pryer, Oliver J Sutton
An 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...
2017: Numerische Mathematik
https://read.qxmd.com/read/29151621/error-analysis-for-discretizations-of-parabolic-problems-using-continuous-finite-elements-in-time-and-mixed-finite-elements-in-space
#24
JOURNAL ARTICLE
Markus Bause, Florin A Radu, Uwe Köcher
Variational time discretization schemes are getting of increasing importance for the accurate numerical approximation of transient phenomena. The applicability and value of mixed finite element methods in space for simulating transport processes have been demonstrated in a wide class of works. We consider a family of continuous Galerkin-Petrov time discretization schemes that is combined with a mixed finite element approximation of the spatial variables. The existence and uniqueness of the semidiscrete approximation and of the fully discrete solution are established...
2017: Numerische Mathematik
https://read.qxmd.com/read/29081544/the-tdnns-method-for-reissner-mindlin-plates
#25
JOURNAL ARTICLE
Astrid S Pechstein, Joachim Schöberl
A new family of locking-free finite elements for shear deformable Reissner-Mindlin plates is presented. The elements are based on the "tangential-displacement normal-normal-stress" formulation of elasticity. In this formulation, the bending moments are treated as separate unknowns. The degrees of freedom for the plate element are the nodal values of the deflection, tangential components of the rotations and normal-normal components of the bending strain. Contrary to other plate bending elements, no special treatment for the shear term such as reduced integration is necessary...
2017: Numerische Mathematik
https://read.qxmd.com/read/28615749/optimal-convergence-for-adaptive-iga-boundary-element-methods-for-weakly-singular-integral-equations
#26
JOURNAL ARTICLE
Michael Feischl, Gregor Gantner, Alexander Haberl, Dirk Praetorius
In a recent work (Feischl et al. in Eng Anal Bound Elem 62:141-153, 2016), we analyzed a weighted-residual error estimator for isogeometric boundary element methods in 2D and proposed an adaptive algorithm which steers the local mesh-refinement of the underlying partition as well as the multiplicity of the knots. In the present work, we give a mathematical proof that this algorithm leads to convergence even with optimal algebraic rates. Technical contributions include a novel mesh-size function which also monitors the knot multiplicity as well as inverse estimates for NURBS in fractional-order Sobolev norms...
2017: Numerische Mathematik
https://read.qxmd.com/read/28615748/discontinuous-galerkin-methods-for-nonlinear-scalar-hyperbolic-conservation-laws-divided-difference-estimates-and-accuracy-enhancement
#27
JOURNAL ARTICLE
Xiong Meng, Jennifer K Ryan
In this paper, an analysis of the accuracy-enhancement for the discontinuous Galerkin (DG) method applied to one-dimensional scalar nonlinear hyperbolic conservation laws is carried out. This requires analyzing the divided difference of the errors for the DG solution. We therefore first prove that the [Formula: see text]-th order [Formula: see text] divided difference of the DG error in the [Formula: see text] norm is of order [Formula: see text] when upwind fluxes are used, under the condition that [Formula: see text] possesses a uniform positive lower bound...
2017: Numerische Mathematik
https://read.qxmd.com/read/28615747/scattered-manifold-valued-data-approximation
#28
JOURNAL ARTICLE
Philipp Grohs, Markus Sprecher, Thomas Yu
We consider the problem of approximating a function f from an Euclidean domain to a manifold M by scattered samples [Formula: see text], where the data sites [Formula: see text] are assumed to be locally close but can otherwise be far apart points scattered throughout the domain. We introduce a natural approximant based on combining the moving least square method and the Karcher mean. We prove that the proposed approximant inherits the accuracy order and the smoothness from its linear counterpart. The analysis also tells us that the use of Karcher's mean (dependent on a Riemannian metric and the associated exponential map) is inessential and one can replace it by a more general notion of 'center of mass' based on a general retraction on the manifold...
2017: Numerische Mathematik
https://read.qxmd.com/read/28615746/local-two-sided-bounds-for-eigenvalues-of-self-adjoint-operators
#29
JOURNAL ARTICLE
G R Barrenechea, L Boulton, N Boussaïd
We examine the equivalence between an extension of the Lehmann-Maehly-Goerisch method developed a few years ago by Zimmermann and Mertins, and a geometrically motivated method developed more recently by Davies and Plum. We establish a general framework which allows sharpening various previously known results in these two settings and determine explicit convergence estimates for both methods. We demonstrate the applicability of the method of Zimmermann and Mertins by means of numerical tests on the resonant cavity problem...
2017: Numerische Mathematik
https://read.qxmd.com/read/28615745/on-the-interconnection-between-the-higher-order-singular-values-of-real-tensors
#30
JOURNAL ARTICLE
Wolfgang Hackbusch, André Uschmajew
A higher-order tensor allows several possible matricizations (reshapes into matrices). The simultaneous decay of singular values of such matricizations has crucial implications on the low-rank approximability of the tensor via higher-order singular value decomposition. It is therefore an interesting question which simultaneous properties the singular values of different tensor matricizations actually can have, but it has not received the deserved attention so far. In this paper, preliminary investigations in this direction are conducted...
2017: Numerische Mathematik
https://read.qxmd.com/read/28615744/regularity-of-non-stationary-subdivision-a-matrix-approach
#31
JOURNAL ARTICLE
M Charina, C Conti, N Guglielmi, V Protasov
In this paper, we study scalar multivariate non-stationary subdivision schemes with integer dilation matrix M and present a unifying, general approach for checking their convergence and for determining their Hölder regularity (latter in the case [Formula: see text]). The combination of the concepts of asymptotic similarity and approximate sum rules allows us to link stationary and non-stationary settings and to employ recent advances in methods for exact computation of the joint spectral radius. As an application, we prove a recent conjecture by Dyn et al...
2017: Numerische Mathematik
https://read.qxmd.com/read/28615743/edge-based-nonlinear-diffusion-for-finite-element-approximations-of-convection-diffusion-equations-and-its-relation-to-algebraic-flux-correction-schemes
#32
JOURNAL ARTICLE
Gabriel R Barrenechea, Erik Burman, Fotini Karakatsani
For the case of approximation of convection-diffusion equations using piecewise affine continuous finite elements a new edge-based nonlinear diffusion operator is proposed that makes the scheme satisfy a discrete maximum principle. The diffusion operator is shown to be Lipschitz continuous and linearity preserving. Using these properties we provide a full stability and error analysis, which, in the diffusion dominated regime, shows existence, uniqueness and optimal convergence. Then the algebraic flux correction method is recalled and we show that the present method can be interpreted as an algebraic flux correction method for a particular definition of the flux limiters...
2017: Numerische Mathematik
https://read.qxmd.com/read/28615742/restarting-iterative-projection-methods-for-hermitian-nonlinear-eigenvalue-problems-with-minmax-property
#33
JOURNAL ARTICLE
Marta M Betcke, Heinrich Voss
In this work we present a new restart technique for iterative projection methods for nonlinear eigenvalue problems admitting minmax characterization of their eigenvalues. Our technique makes use of the minmax induced local enumeration of the eigenvalues in the inner iteration. In contrast to global numbering which requires including all the previously computed eigenvectors in the search subspace, the proposed local numbering only requires a presence of one eigenvector in the search subspace. This effectively eliminates the search subspace growth and therewith the super-linear increase of the computational costs if a large number of eigenvalues or eigenvalues in the interior of the spectrum are to be computed...
2017: Numerische Mathematik
https://read.qxmd.com/read/28615741/runge-kutta-time-semidiscretizations-of-semilinear-pdes-with-non-smooth-data
#34
JOURNAL ARTICLE
Claudia Wulff, Chris Evans
We study semilinear evolution equations [Formula: see text] posed on a Hilbert space [Formula: see text], where A is normal and generates a strongly continuous semigroup, B is a smooth nonlinearity from [Formula: see text] to itself, and [Formula: see text], [Formula: see text], [Formula: see text]. In particular the one-dimensional semilinear wave equation and nonlinear Schrödinger equation with periodic, Neumann and Dirichlet boundary conditions fit into this framework. We discretize the evolution equation with an A-stable Runge-Kutta method in time, retaining continuous space, and prove convergence of order [Formula: see text] for non-smooth initial data [Formula: see text], where [Formula: see text], for a method of classical order p, extending a result by Brenner and Thomée for linear systems...
2016: Numerische Mathematik
https://read.qxmd.com/read/28615740/backward-error-analysis-of-the-shift-and-invert-arnoldi-algorithm
#35
JOURNAL ARTICLE
Christian Schröder, Leo Taslaman
We perform a backward error analysis of the inexact shift-and-invert Arnoldi algorithm. We consider inexactness in the solution of the arising linear systems, as well as in the orthonormalization steps, and take the non-orthonormality of the computed Krylov basis into account. We show that the computed basis and Hessenberg matrix satisfy an exact shift-and-invert Krylov relation for a perturbed matrix, and we give bounds for the perturbation. We show that the shift-and-invert Arnoldi algorithm is backward stable if the condition number of the small Hessenberg matrix is not too large...
2016: Numerische Mathematik
https://read.qxmd.com/read/28615739/mixed-finite-elements-for-global-tide-models
#36
JOURNAL ARTICLE
Colin J Cotter, Robert C Kirby
We study mixed finite element methods for the linearized rotating shallow water equations with linear drag and forcing terms. By means of a strong energy estimate for an equivalent second-order formulation for the linearized momentum, we prove long-time stability of the system without energy accumulation-the geotryptic state. A priori error estimates for the linearized momentum and free surface elevation are given in [Formula: see text] as well as for the time derivative and divergence of the linearized momentum...
2016: Numerische Mathematik
https://read.qxmd.com/read/28615738/a-massively-parallel-nonoverlapping-additive-schwarz-method-for-discontinuous-galerkin-discretization-of-elliptic-problems
#37
JOURNAL ARTICLE
Maksymilian Dryja, Piotr Krzyżanowski
A second order elliptic problem with discontinuous coefficient in 2-D or 3-D is considered. The problem is discretized by a symmetric weighted interior penalty discontinuous Galerkin finite element method with nonmatching simplicial elements and piecewise linear functions. The resulting discrete problem is solved by a two-level additive Schwarz method with a relatively coarse grid and with local solves restricted to subdomains which can be as small as single element. In this way the method has a potential for a very high level of fine grained parallelism...
2016: Numerische Mathematik
https://read.qxmd.com/read/28603298/a-stable-numerical-method-for-the-dynamics-of-fluidic-membranes
#38
JOURNAL ARTICLE
John W Barrett, Harald Garcke, Robert Nürnberg
We develop a finite element scheme to approximate the dynamics of two and three dimensional fluidic membranes in Navier-Stokes flow. Local inextensibility of the membrane is ensured by solving a tangential Navier-Stokes equation, taking surface viscosity effects of Boussinesq-Scriven type into account. In our approach the bulk and surface degrees of freedom are discretized independently, which leads to an unfitted finite element approximation of the underlying free boundary problem. Bending elastic forces resulting from an elastic membrane energy are discretized using an approximation introduced by Dziuk (Numer Math 111:55-80, 2008)...
2016: Numerische Mathematik
https://read.qxmd.com/read/28615737/automatic-integration-using-asymptotically-optimal-adaptive-simpson-quadrature
#39
JOURNAL ARTICLE
Leszek Plaskota
We present a novel theoretical approach to the analysis of adaptive quadratures and adaptive Simpson quadratures in particular which leads to the construction of a new algorithm for automatic integration. For a given function [Formula: see text] with [Formula: see text] and possible endpoint singularities the algorithm produces an approximation to [Formula: see text] within a given [Formula: see text] asymptotically as [Formula: see text]. Moreover, it is optimal among all adaptive Simpson quadratures, i.e...
2015: Numerische Mathematik
https://read.qxmd.com/read/28615736/an-analysis-of-the-rayleigh-stokes-problem-for-a-generalized-second-grade-fluid
#40
JOURNAL ARTICLE
Emilia Bazhlekova, Bangti Jin, Raytcho Lazarov, Zhi Zhou
We study the Rayleigh-Stokes problem for a generalized second-grade fluid which involves a Riemann-Liouville fractional derivative in time, and present an analysis of the problem in the continuous, space semidiscrete and fully discrete formulations. We establish the Sobolev regularity of the homogeneous problem for both smooth and nonsmooth initial data [Formula: see text], including [Formula: see text]. A space semidiscrete Galerkin scheme using continuous piecewise linear finite elements is developed, and optimal with respect to initial data regularity error estimates for the finite element approximations are derived...
2015: Numerische Mathematik
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