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Numerische Mathematik

Monika Eisenmann, Eskil Hansen
Domain decomposition based time integrators allow the usage of parallel and distributed hardware, making them well-suited for the temporal discretization of parabolic systems. In this study, a rigours convergence analysis is given for such integrators without assuming any restrictive regularity on the solutions or the domains. The analysis is conducted by first deriving a new variational framework for the domain decomposition, which is applicable to the two standard degenerate examples. That is, the p -Laplace and the porous medium type vector fields...
2018: Numerische Mathematik
Markus Faustmann, Jens Markus Melenk
The local behavior of the lowest order boundary element method on quasi-uniform meshes for Symm's integral equation and the stabilized hyper-singular integral equation on polygonal/polyhedral Lipschitz domains is analyzed. We prove local a priori estimates in <mml:math xmlns:mml=""> <mml:msup> <mml:mi>L</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:math> for Symm's integral equation and in <mml:math xmlns:mml="http://www.w3...
2018: Numerische Mathematik
Ivan G Graham, Frances Y Kuo, Dirk Nuyens, Rob Scheichl, Ian H Sloan
In a previous paper (Graham et al. in J Comput Phys 230:3668-3694, 2011), the authors proposed a new practical method for computing expected values of functionals of solutions for certain classes of elliptic partial differential equations with random coefficients. This method was based on combining quasi-Monte Carlo (QMC) methods for computing the expected values with circulant embedding methods for sampling the random field on a regular grid. It was found capable of handling fluid flow problems in random heterogeneous media with high stochastic dimension, but no convergence theory was provided...
2018: Numerische Mathematik
Barbara Kaltenbacher, Mario Luiz Previatti de Souza
In this paper we consider the iteratively regularized Gauss-Newton method (IRGNM) in its classical Tikhonov version as well as two further-Ivanov type and Morozov type-versions. In these two alternative versions, regularization is achieved by imposing bounds on the solution or by minimizing some regularization functional under a constraint on the data misfit, respectively. We do so in a general Banach space setting and under a tangential cone condition, while convergence (without source conditions, thus without rates) has so far only been proven under stronger restrictions on the nonlinearity of the operator and/or on the spaces...
2018: Numerische Mathematik
Martin Campos Pinto, José A Carrillo, Frédérique Charles, Young-Pil Choi
We study a linearly transformed particle method for the aggregation equation with smooth or singular interaction forces. For the smooth interaction forces, we provide convergence estimates in <mml:math xmlns:mml=""> <mml:msup> <mml:mi>L</mml:mi> <mml:mn>1</mml:mn> </mml:msup> </mml:math> and <mml:math xmlns:mml=""> <mml:msup> <mml:mi>L</mml:mi> <mml:mi>∞</mml:mi> </mml:msup> </mml:math> norms depending on the regularity of the initial data...
2018: Numerische Mathematik
Nikolaos Rekatsinas, Rob Stevenson
In this paper, it is shown that any well-posed 2nd order PDE can be reformulated as a well-posed first order least squares system. This system will be solved by an adaptive wavelet solver in optimal computational complexity. The applications that are considered are second order elliptic PDEs with general inhomogeneous boundary conditions, and the stationary Navier-Stokes equations.
2018: Numerische Mathematik
Klaus Deckelnick, Vanessa Styles
In this paper we analyze a fully discrete numerical scheme for solving a parabolic PDE on a moving surface. The method is based on a diffuse interface approach that involves a level set description of the moving surface. Under suitable conditions on the spatial grid size, the time step and the interface width we obtain stability and error bounds with respect to natural norms. Furthermore, we present test calculations that confirm our analysis.
2018: Numerische Mathematik
Shinji Ito, Yuji Nakatsukasa
A common way of finding the poles of a meromorphic function f in a domain, where an explicit expression of f is unknown but f can be evaluated at any given z , is to interpolate f by a rational function <mml:math xmlns:mml=""> <mml:mfrac> <mml:mi>p</mml:mi> <mml:mi>q</mml:mi> </mml:mfrac> </mml:math> such that <mml:math xmlns:mml=""> <mml:mrow> <mml:mi>r</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:msub> <mml:mi>γ</mml:mi> <mml:mi>i</mml:mi> </mml:msub> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>=</mml:mo> <mml:mi>f</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:msub> <mml:mi>γ</mml:mi> <mml:mi>i</mml:mi> </mml:msub> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> at prescribed sample points <mml:math xmlns:mml="http://www...
2018: Numerische Mathematik
Erik Burman, Lauri Oksanen
We consider data assimilation for the heat equation using a finite element space semi-discretization. The approach is optimization based, but the design of regularization operators and parameters rely on techniques from the theory of stabilized finite elements. The space semi-discretized system is shown to admit a unique solution. Combining sharp estimates of the numerical stability of the discrete scheme and conditional stability estimates of the ill-posed continuous pde-model we then derive error estimates that reflect the approximation order of the finite element space and the stability of the continuous model...
2018: Numerische Mathematik
Astrid S Pechstein, Joachim Schöberl
The tangential-displacement normal-normal-stress (TDNNS) method is a finite element method for mixed elasticity. As the name suggests, the tangential component of the displacement vector as well as the normal-normal component of the stress are the degrees of freedom of the finite elements. The TDNNS method was shown to converge of optimal order, and to be robust with respect to shear and volume locking. However, the method is slightly nonconforming, and an analysis with respect to the natural norms of the arising spaces was still missing...
2018: Numerische Mathematik
Anton Arnold, Claudia Negulescu
This paper is concerned with a 1D Schrödinger scattering problem involving both oscillatory and evanescent regimes, separated by jump discontinuities in the potential function, to avoid "turning points". We derive a non-overlapping domain decomposition method to split the original problem into sub-problems on these regions, both for the continuous and afterwards for the discrete problem. Further, a hybrid WKB-based numerical method is designed for its efficient and accurate solution in the semi-classical limit: a WKB-marching method for the oscillatory regions and a FEM with WKB-basis functions in the evanescent regions...
2018: Numerische Mathematik
John W Pearson, Jennifer Pestana, David J Silvester
This paper is concerned with the implementation of efficient solution algorithms for elliptic problems with constraints. We establish theory which shows that including a simple scaling within well-established block diagonal preconditioners for Stokes problems can result in significantly faster convergence when applying the preconditioned MINRES method. The codes used in the numerical studies are available online.
2018: Numerische Mathematik
Peter Benner, Zvonimir Bujanović, Patrick Kürschner, Jens Saak
This paper introduces a new algorithm for solving large-scale continuous-time algebraic Riccati equations (CARE). The advantage of the new algorithm is in its immediate and efficient low-rank formulation, which is a generalization of the Cholesky-factored variant of the Lyapunov ADI method. We discuss important implementation aspects of the algorithm, such as reducing the use of complex arithmetic and shift selection strategies. We show that there is a very tight relation between the new algorithm and three other algorithms for CARE previously known in the literature-all of these seemingly different methods in fact produce exactly the same iterates when used with the same parameters: they are algorithmically different descriptions of the same approximation sequence to the Riccati solution...
2018: Numerische Mathematik
Gunther Leobacher, Michaela Szölgyenyi
We prove strong convergence of order [Formula: see text] for arbitrarily small [Formula: see text] of the Euler-Maruyama method for multidimensional stochastic differential equations (SDEs) with discontinuous drift and degenerate diffusion coefficient. The proof is based on estimating the difference between the Euler-Maruyama scheme and another numerical method, which is constructed by applying the Euler-Maruyama scheme to a transformation of the SDE we aim to solve.
2018: Numerische Mathematik
Axel Målqvist, Anna Persson
We use the local orthogonal decomposition technique introduced in Målqvist and Peterseim (Math Comput 83(290):2583-2603, 2014) to derive a generalized finite element method for linear and semilinear parabolic equations with spatial multiscale coefficients. We consider nonsmooth initial data and a backward Euler scheme for the temporal discretization. Optimal order convergence rate, depending only on the contrast, but not on the variations of the coefficients, is proven in the [Formula: see text]-norm. We present numerical examples, which confirm our theoretical findings...
2018: Numerische Mathematik
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
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
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
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
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
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