Michael Karkulik

Michael Karkulik, David Pavlicek, Dirk Praetorius

Newest vertex bisection (NVB) is a popular local mesh-refinement strategy for regular triangulations which consist of simplices. For the 2D case, we prove that the mesh-closure step of NVB, which preserves regularity of the triangulation, is quasi-optimal and that the corresponding L2-projection onto lowest-order Courant finite elements (P1-FEM) is always H1-stable. Throughout, no additional assumptions on the initial triangulation are imposed. Our analysis thus improves results of Binev, Dahmen & DeVore (Numer. Math. 97, 2004), Carstensen (Constr. Approx. 20, 2004), and Stevenson (Math. Comp. 77, 2008) in the sense that all assumptions of their theorems are removed. Consequently, our results relax the requirements under which adaptive finite element schemes can be mathematically guaranteed to convergence with quasi-optimal rates.

Markus Aurada, Michael Feischl, Thomas Führer et al.

We consider a (possibly) nonlinear interface problem in 2D and 3D, which is solved by use of various adaptive FEM-BEM coupling strategies, namely the Johnson-Nédélec coupling, the Bielak-MacCamy coupling, and Costabel's symmetric coupling. We provide a framework to prove that the continuous as well as the discrete Galerkin solutions of these coupling methods additionally solve an appropriate operator equation with a Lipschitz continuous and strongly monotone operator. Therefore, the coupling formulations are well-defined, and the Galerkin solutions are quasi-optimal in the sense of a Céa-type lemma. For the respective Galerkin discretizations with lowest-order polynomials, we provide reliable residual-based error estimators. Together with an estimator reduction property, we prove convergence of the adaptive FEM-BEM coupling methods. A key point for the proof of the estimator reduction are novel inverse-type estimates for the involved boundary integral operators which are advertized. Numerical experiments conclude the work and compare performance and effectivity of the three adaptive coupling procedures in the presence of generic singularities.

Michael Feischl, Thomas Führer, Michael Karkulik et al.

We consider symmetric as well as non-symmetric coupling formulations of FEM and BEM in the frame of nonlinear elasticity problems. In particular, the Johnson-Nédélec coupling is analyzed. We prove that these coupling formulations are well-posed and allow for unique Galerkin solutions if standard discretizations by piecewise polynomials are employed. Unlike prior works, our analysis does neither rely on an interior Dirichlet boundary to tackle the rigid body motions nor on any assumption on the mesh-size of the discretization used.

Markus Aurada, Michael Feischl, Thomas Führer et al.

We prove inverse-type estimates for the four classical boundary integral operators associated with the Laplace operator. These estimates are used to show convergence of an h-adaptive algorithm for the coupling of a finite element method with a boundary element method which is driven by a weighted residual error estimator.

Norbert Heuer, Michael Karkulik

Generalizing the framework of an ultra-weak formulation for a hypersingular integral equation on closed polygons in [N. Heuer, F. Pinochet, arXiv 1309.1697 (to appear in SIAM J. Numer. Anal.)], we study the case of a hypersingular integral equation on open and closed polyhedral surfaces. We develop a general ultra-weak setting in fractional-order Sobolev spaces and prove its well-posedness and equivalence with the traditional formulation. Based on the ultra-weak formulation, we establish a discontinuous Petrov-Galerkin method with optimal test functions and prove its quasi-optimal convergence in related Sobolev norms. For closed surfaces, this general result implies quasi-optimal convergence in the L^2-norm. Some numerical experiments confirm expected convergence rates.

Thomas Führer, Norbert Heuer, Michael Karkulik et al.

We propose and analyze a discretization scheme that combines the discontinuous Petrov-Galerkin and finite element methods. The underlying model problem is of general diffusion-advection-reaction type on bounded domains, with decomposition into two sub-domains. We propose a heterogeneous variational formulation that is of the ultra-weak (Petrov-Galerkin) form with broken test space in one part, and of Bubnov-Galerkin form in the other. A standard discretization with conforming approximation spaces and appropriate test spaces (optimal test functions for the ultra-weak part and standard test functions for the Bubnov-Galerkin part) gives rise to a coupled DPG-FEM scheme. We prove its well-posedness and quasi-optimal convergence. Numerical results confirm expected convergence orders.

Michael Karkulik

We consider the finite element discretization of an optimal Dirichlet boundary control problem for the Laplacian, where the control is considered in $H^{1/2}(Γ)$. To avoid computing the latter norm numerically, we realize it using the $H^{1}(Ω)$ norm of the harmonic extension of the control. We propose a mixed finite element discretization, where the harmonicity of the solution is included by a Lagrangian multiplier. In the case of convex polygonal domains, optimal error estimates in the $H^1$ and $L^2$ norm are proven. We also consider and analyze the case of control constrained problems.

Michael Karkulik

We consider initial/boundary value problems for time-fractional parabolic PDE of order $0<α<1$ with Caputo fractional derivative (also called fractional diffusion equations in the literature). We prove well-posedness of corresponding variational formulations based entirely on fractional Sobolev-Bochner spaces, and clarify the question of possible choices of the initial value.

Thomas Führer, Michael Karkulik

We provide new insights into the a priori theory for a time-stepping scheme based on least-squares finite element methods for parabolic first-order systems. The elliptic part of the problem is of general reaction-convection-diffusion type. The new ingredient in the analysis is an elliptic projection operator defined via a non-symmetric bilinear form, although the main bilinear form corresponding to the least-squares functional is symmetric. This new operator allows to prove optimal error estimates in the natural norm associated to the problem and, under additional regularity assumptions, in the $L^2$ norm. Numerical experiments are presented which confirm our theoretical findings.

Michael Karkulik, Jens Markus Melenk

The integral version of the fractional Laplacian on a bounded domain is discretized by a Galerkin approximation based on piecewise linear functions on a quasi-uniform mesh. We show that the inverse of the associated stiffness matrix can be approximated by blockwise low-rank matrices at an exponential rate in the block rank.

Norbert Heuer, Michael Karkulik

We present and analyze a discontinuous Petrov-Galerkin method with optimal test functions for a reaction-dominated diffusion problem in two and three space dimensions. We start with an ultra-weak formulation that comprises parameters $α$, $β$ to allow for general $\varepsilon$-dependent weightings of three field variables ($\varepsilon$ being the small diffusion parameter). Specific values of $α$ and $β$ imply robustness of the method, that is, a quasi-optimal error estimate with a constant that is independent of $\varepsilon$. Moreover, these values lead to a norm for the field variables that is known to be balanced in $\varepsilon$ for model problems with typical boundary layers. Several numerical examples underline our theoretical estimates and reveal stability of approximations even for very small $\varepsilon$.

Michael Karkulik, Jens Markus Melenk

We develop a regularization operator based on smoothing on a locally defined length scale. This operator is defined on $L_1$ and has approximation properties that are given by the local regularity of the function it is applied to and the local length scale. Additionally, the regularized function satisfies inverse estimates commensurate with the approximation orders. By combining this operator with a classical hp-interpolation operator, we obtain an hp-Clément type quasi-interpolation operator, i.e., an operator that requires minimal smoothness of the function to be approximated but has the expected approximation properties in terms of the local mesh size and polynomial degree. As a second application, we consider residual error estimates in hp-boundary element methods that are explicit in the local mesh size and the local approximation order.

Thomas Führer, Norbert Heuer, Michael Karkulik

We develop and analyze strategies to couple the discontinuous Petrov-Galerkin method with optimal test functions to (i) least-squares boundary elements and (ii) various variants of standard Galerkin boundary elements. Essential feature of our methods is that, despite the use of boundary integral equations, optimal test functions have to be computed only locally. We apply our findings to a standard transmission problem in full space and present numerical experiments to validate our theory.

Vincent J. Ervin, Thomas Führer, Norbert Heuer et al.

We develop an ultra-weak variational formulation of a fractional advection diffusion problem in one space dimension and prove its well-posedness. Based on this formulation, we define a DPG approximation with optimal test functions and show its quasi-optimal convergence. Numerical experiments confirm expected convergence properties, for uniform and adaptively refined meshes.

Markus Aurada, Michael Feischl, Thomas Führer et al.

We prove local inverse-type estimates for the four non-local boundary integral operators associated with the Laplace operator on a bounded d-dimensional Lipschitz domain Omega for d >= 2 with piecewise smooth boundary. For piecewise polynomial ansatz spaces and d = 2 or 3, the inverse estimates are explicit in both the local mesh width and the approximation order. An application to efficiency estimates in a posteriori error estimation in boundary element methods is given.

Michael Karkulik, Carl-Martin Pfeiler, Dirk Praetorius

We merge and extend recent results which prove the H1-stability of the L2-orthogonal projection onto standard finite element spaces, provided that the underlying simplicial triangulation is appropriately graded. For lowest-order Courant finite elements S1(T) in Rd with d>=2, we prove that such a grading is always ensured for adaptive meshes generated by newest vertex bisection. For higher-order finite elements Sp(T) with p>=1, we extend existing bounds on the polynomial degree with a computer-assisted proof. We also consider L2-orthogonal projections onto certain subspaces of Sp(T) which incorporate zero Dirichlet boundary conditions resp. an integral mean zero property.

Norbert Heuer, Michael Karkulik

We propose and analyze a numerical method to solve an elliptic transmission problem in full space. The method consists of a variational formulation involving standard boundary integral operators on the coupling interface and an ultra-weak formulation in the interior. To guarantee the discrete inf-sup condition, the system is discretized by the DPG method with optimal test functions. We prove that principal unknowns are approximated quasi-optimally. Numerical experiments for problems with smooth and singular solutions confirm optimal convergence orders.