arXiv subjects

Gregor Gantner

Publications and source records attributed to Gregor Gantner.

36 records · Page 2Linked to original sources

arXiv2022

Adaptive BEM for elliptic PDE systems, part II: Isogeometric analysis with hierarchical B-splines for weakly-singular integral equations

We formulate and analyze an adaptive algorithm for isogeometric analysis with hierarchical B-splines for weakly-singular boundary integral equations. We prove that the employed weighted-residual error estimator is reliable and converges at optimal algebraic rate. Numerical experiments with isogeometric boundary elements for the 3D Poisson problem confirm the theoretical results, which also cover general elliptic systems like linear elasticity.

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arXiv2022

A well-posed First Order System Least Squares formulation of the instationary Stokes equations

In this paper, a well-posed simultaneous space-time First Order System Least Squares formulation is constructed of the instationary incompressible Stokes equations with slip boundary conditions. As a consequence of this well-posedness, the minimization over any conforming triple of finite element spaces for velocities, pressure and stress tensor gives a quasi-best approximation from that triple. The formulation is practical in the sense that all norms in the least squares functional can be efficiently evaluated. Being of least squares type, the formulation comes with an efficient and reliable a posteriori error estimator. In addition, a priori error estimates are derived, and numerical results are presented.

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arXiv2021

Efficient numerical approximation of a non-regular Fokker--Planck equation associated with first-passage time distributions

In neuroscience, the distribution of a decision time is modelled by means of a one-dimensional Fokker--Planck equation with time-dependent boundaries and space-time-dependent drift. Efficient approximation of the solution to this equation is required, e.g., for model evaluation and parameter fitting. However, the prescribed boundary conditions lead to a strong singularity and thus to slow convergence of numerical approximations. In this article we demonstrate that the solution can be related to the solution of a parabolic PDE on a rectangular space-time domain with homogeneous initial and boundary conditions by transformation and subtraction of a known function. We verify that the solution of the new PDE is indeed more regular than the solution of the original PDE and proceed to discretize the new PDE using a space-time minimal residual method. We also demonstrate that the solution depends analytically on the parameters determining the boundaries as well as the drift. This justifies the use of a sparse tensor product interpolation method to approximate the PDE solution for various parameter ranges. The predicted convergence rates of the minimal residual method and that of the interpolation method are supported by numerical simulations.

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arXiv2021

Adaptive space-time BEM for the heat equation

We consider the space-time boundary element method (BEM) for the heat equation with prescribed initial and Dirichlet data. We propose a residual-type a posteriori error estimator that is a lower bound and, up to weighted $L_2$-norms of the residual, also an upper bound for the unknown BEM error. The possibly locally refined meshes are assumed to be prismatic, i.e., their elements are tensor-products $J\times K$ of elements in time $J$ and space $K$. While the results do not depend on the local aspect ratio between time and space, assuming the scaling $|J| \eqsim {\rm diam}(K)^2$ for all elements and using Galerkin BEM, the estimator is shown to be efficient and reliable without the additional $L_2$-terms. In the considered numerical experiments on two-dimensional domains in space, the estimator seems to be equivalent to the error, independently of these assumptions. In particular for adaptive anisotropic refinement, both converge with the best possible convergence rate.

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arXiv2021

Further results on a space-time FOSLS formulation of parabolic PDEs

In [2019, Space-time least-squares finite elements for parabolic equations, arXiv:1911.01942] by Führer& Karkulik, well-posedness of a space-time First-Order System Least-Squares formulation of the heat equation was proven. In the present work, this result is generalized to general second order parabolic PDEs with possibly inhomogenoeus boundary conditions, and plain convergence of a standard adaptive finite element method driven by the least-squares estimator is demonstrated. The proof of the latter easily extends to a large class of least-squares formulations.

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arXiv2020

Plain convergence of adaptive algorithms without exploiting reliability and efficiency

We consider h-adaptive algorithms in the context of the finite element method (FEM) and the boundary element method (BEM). Under quite general assumptions on the building blocks SOLVE, ESTIMATE, MARK, and REFINE of such algorithms, we prove plain convergence in the sense that the adaptive algorithm drives the underlying a posteriori error estimator to zero. Unlike available results in the literature, our analysis avoids the use of any reliability and efficiency estimate, but only relies on structural properties of the estimator, namely stability on non-refined elements and reduction on refined elements. In particular, the new framework thus covers also problems involving non-local operators like the fractional Laplacian or boundary integral equations, where (discrete) efficiency is (currently) not available.

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arXiv2020

Adaptive BEM for elliptic PDE systems, Part I: Abstract framework for weakly-singular integral equations

In the present work, we consider weakly-singular integral equations arising from linear second-order strongly-elliptic PDE systems with constant coefficients, including, e.g., linear elasticity. We introduce a general framework for optimal convergence of adaptive Galerkin BEM. We identify certain abstract properties for the underlying meshes, the corresponding mesh-refinement strategy, and the ansatz spaces that guarantee convergence at optimal algebraic rate of an adaptive algorithm driven by the weighted-residual error. These properties are satisfied, e.g., for discontinuous piecewise polynomials on simplicial meshes as well as certain ansatz spaces used for isogeometric analysis. Technical contributions include local inverse estimates for the (non-local) boundary integral operators associated to the PDE system.

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arXiv2020

Rate optimality of adaptive finite element methods with respect to the overall computational costs

We consider adaptive finite element methods for second-order elliptic PDEs, where the arising discrete systems are not solved exactly. For contractive iterative solvers, we formulate an adaptive algorithm which monitors and steers the adaptive mesh-refinement as well as the inexact solution of the arising discrete systems. We prove that the proposed strategy leads to linear convergence with optimal algebraic rates. Unlike prior works, however, we focus on convergence rates with respect to the overall computational costs. In explicit terms, the proposed adaptive strategy thus guarantees quasi-optimal computational time. In particular, our analysis covers linear problems, where the linear systems are solved by an optimally preconditioned CG method as well as nonlinear problems with strongly monotone nonlinearity which are linearized by the so-called Zarantonello iteration.

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arXiv2020

Adaptive IGAFEM with optimal convergence rates: T-splines

We consider an adaptive algorithm for finite element methods for the isogeometric analysis (IGAFEM) of elliptic (possibly non-symmetric) second-order partial differential equations. We employ analysis-suitable T-splines of arbitrary odd degree on T-meshes generated by the refinement strategy of [Morgenstern, Peterseim, Comput. Aided Geom. Design 34 (2015)] in 2D and [Morgenstern, SIAM J. Numer. Anal. 54 (2016)] in 3D. Adaptivity is driven by some weighted residual a posteriori error estimator. We prove linear convergence of the error estimator (which is equivalent to the sum of energy error plus data oscillations) with optimal algebraic rates with respect to the number of elements of the underlying mesh.

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arXiv2019

Adaptive Uzawa algorithm for the Stokes equation

Based on the Uzawa algorithm, we consider an adaptive finite element method for the Stokes system. We prove linear convergence with optimal algebraic rates for the residual estimator (which is equivalent to the total error), if the arising linear systems are solved iteratively, e.g., by PCG. Our analysis avoids the use of discrete efficiency of the estimator. Unlike prior work, our adaptive Uzawa algorithm can thus avoid to discretize the given data and does not rely on an interior node property for the refinement.

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arXiv2019

Optimal additive Schwarz preconditioning for adaptive 2D IGA boundary element methods

We define and analyze (local) multilevel diagonal preconditioners for isogeometric boundary elements on locally refined meshes in two dimensions. Hypersingular and weakly-singular integral equations are considered. We prove that the condition number of the preconditioned systems of linear equations is independent of the mesh-size and the refinement level. Therefore, the computational complexity, when using appropriate iterative solvers, is optimal. Our analysis is carried out for closed and open boundaries and numerical examples confirm our theoretical results.

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arXiv2019

Adaptive isogeometric boundary element methods with local smoothness control

In the frame of isogeometric analysis, we consider a Galerkin boundary element discretization of the hyper-singular integral equation associated with the 2D Laplacian. We propose and analyze an adaptive algorithm which locally refines the boundary partition and, moreover, steers the smoothness of the NURBS ansatz functions across elements. In particular and unlike prior work, the algorithm can increase and decrease the local smoothness properties and hence exploits the full potential of isogeometric analysis. We prove that the new adaptive strategy leads to linear convergence with optimal algebraic rates. Numerical experiments confirm the theoretical results. A short appendix comments on analogous results for the weakly-singular integral equation.

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arXiv2019

Optimal convergence behavior of adaptive FEM driven by simple (h-h/2)-type error estimators

For some Poisson-type model problem, we prove that adaptive FEM driven by the (h-h/2)-type error estimators from [Ferraz-Leite, Ortner, Praetorius, Numer. Math. 116 (2010)] leads to convergence with optimal algebraic convergence rates. Besides the implementational simplicity, another striking feature of these estimators is that they can provide guaranteed lower bounds for the energy error with known efficiency constant 1.

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arXiv2017

Rate optimal adaptive FEM with inexact solver for nonlinear operators

We prove convergence with optimal algebraic rates for an adaptive finite element method for nonlinear equations with strongly monotone operator. Unlike prior works, our analysis also includes the iterative and inexact solution of the arising nonlinear systems by means of the Picard iteration. Using nested iteration, we prove, in particular, that the number of of Picard iterations is uniformly bounded in generic cases, and the overall computational cost is (almost) optimal. Numerical experiments confirm the theoretical results.

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arXiv2017

Adaptive IGAFEM with optimal convergence rates: Hierarchical B-splines

We consider an adaptive algorithm for finite element methods for the isogeometric analysis (IGAFEM) of elliptic (possibly non-symmetric) second-order partial differential equations in arbitrary space dimension $d\ge2$. We employ hierarchical B-splines of arbitrary degree and different order of smoothness. We propose a refinement strategy to generate a sequence of locally refined meshes and corresponding discrete solutions. Adaptivity is driven by some weighted residual a posteriori error estimator. We prove linear convergence of the error estimator (resp. the sum of energy error plus data oscillations) with optimal algebraic rates. Numerical experiments underpin the theoretical findings.

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arXiv2015

Optimal convergence for adaptive IGA boundary element methods for weakly-singular integral equations

In a recent work, 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.

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arXiv2015

Adaptive 2D IGA boundary element methods

We derive and discuss a posteriori error estimators for Galerkin and collocation IGA boundary element methods for weakly-singular integral equations of the first-kind in 2D. While recent own work considered the Faermann residual error estimator for Galerkin IGA boundary element methods, the present work focuses more on collocation and weighted- residual error estimators, which provide reliable upper bounds for the energy error. Our analysis allows piecewise smooth parametrizations of the boundary, local mesh-refinement, and related standard piecewise polynomials as well as NURBS. We formulate an adaptive algorithm which steers the local mesh-refinement and the multiplicity of the knots. Numerical experiments show that the proposed adaptive strategy leads to optimal convergence, and related IGA boundary element methods are superior to standard boundary element methods with piecewise polynomials.

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arXiv2014

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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