42 citations · 176 across the 19 of their papers we have counts for
21 papers
A pollution-free ultra-weak FOSLS discretization of the Helmholtz equation
Harald Monsuur, Rob Stevenson
We consider an ultra-weak first order system discretization of the Helmholtz equation. When employing the optimal test norm, the `ideal' method yields the best approximation to the…
A convenient inclusion of inhomogeneous boundary conditions in minimal residual methods
Rob Stevenson
Inhomogeneous essential boundary conditions can be appended to a well-posed PDE to lead to a combined variational formulation. The domain of the corresponding operator is a Sobolev…
Minimal residual methods in negative or fractional Sobolev norms
Harald Monsuur, Rob Stevenson, Johannes Storn
For numerical approximation the reformulation of a PDE as a residual minimisation problem has the advantages that the resulting linear system is symmetric positive definite, and th…
Interpolation Operators for parabolic Problems
Rob Stevenson, Johannes Storn
We introduce interpolation operators with approximation and stability properties suited for parabolic problems in primal and mixed formulations. We derive localized error estimates…
Improved rates for a space-time FOSLS of parabolic PDEs
Gregor Gantner, Rob Stevenson
We consider the first-order system space-time formulation of the heat equation introduced in [Bochev, Gunzburger, Springer, New York (2009)], and analyzed in [Führer, Karkulik, Com…
Applications of a space-time FOSLS formulation for parabolic PDEs
Gregor Gantner, Rob Stevenson
In this work, we show that the space-time first-order system least-squares (FOSLS) formulation [Führer, Karkulik, Comput. Math. Appl. 92 (2021)] for the heat equation and its recen…
Least squares solvers for ill-posed PDEs that are conditionally stable
Wolfgang Dahmen, Harald Monsuur, Rob Stevenson
This paper is concerned with the design and analysis of least squares solvers for ill-posed PDEs that are conditionally stable. The norms and the regularization term used in the le…
A well-posed First Order System Least Squares formulation of the instationary Stokes equations
Gregor Gantner, Rob Stevenson
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 bounda…
Efficient numerical approximation of a non-regular Fokker--Planck equation associated with first-passage time distributions
Udo Boehm, Sonja Cox, Gregor Gantner +1
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…
A weighted POD-reduction approach for parametrized PDE-constrained Optimal Control Problems with random inputs and applications to environmental sciences
Giuseppe Carere, Maria Strazzullo, Francesco Ballarin +2
Reduced basis approximations of Optimal Control Problems (OCPs) governed by steady partial differential equations (PDEs) with random parametric inputs are analyzed and constructed.…
Operator preconditioning: the simplest case
Rob Stevenson, Raymond van Venetië
Using the framework of operator or Calderón preconditioning, uniform preconditioners are constructed for elliptic operators discretized with continuous finite (or boundary) element…
A wavelet-in-time, finite element-in-space adaptive method for parabolic evolution equations
Rob Stevenson, Raymond van Venetië, Jan Westerdiep
In this work, an -linearly converging adaptive solver is constructed for parabolic evolution equations in a simultaneous space-time variational formulation. Exploiting the produ…
Further results on a space-time FOSLS formulation of parabolic PDEs
Gregor Gantner, Rob Stevenson
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 f…
Stability of Galerkin discretizations of a mixed space-time variational formulation of parabolic evolution equations
Rob Stevenson, Jan Westerdiep
We analyze Galerkin discretizations of a new well-posed mixed space-time variational formulation of parabolic PDEs. For suitable pairs of finite element trial spaces, the resulting…
Adaptive Strategies for Transport Equations
W. Dahmen, R. P. Stevenson
This paper is concerned with a posteriori error bounds for linear transport equations and related questions of contriving corresponding adaptive solution strategies in the context…
A quadratic finite element wavelet Riesz basis
Nikolaos Rekatsinas, Rob Stevenson
In this paper, continuous piecewise quadratic finite element wavelets are constructed on general polygons in . The wavelets are stable in for …
A saturation property for the spectral-Galerkin approximation of a Dirichlet problem in a square
Claudio Canuto, Ricardo H. Nochetto, Rob Stevenson +1
Both practice and analysis of adaptive -FEMs and -FEMs raise the question what increment in the current polynomial degree guarantees a -independent reduction of the G…
On p-Robust Saturation for hp-AFEM
Claudio Canuto, Ricardo H. Nochetto, Rob Stevenson +1
We consider the standard adaptive finite element loop SOLVE, ESTIMATE, MARK, REFINE, with ESTIMATE being implemented using the -robust equilibrated flux estimator, and MARK bein…
On the stability of DPG formulations of transport equations
D. Broersen, W. Dahmen, R. P. Stevenson
In this paper we formulate and analyze a Discontinuous Petrov Galerkin formulation of linear transport equations with variable convection fields. We show that a corresponding {\em…
Convergence and Optimality of hp-AFEM
Claudio Canuto, Ricardo H. Nochetto, Rob Stevenson +1
We design and analyze an adaptive -finite element method (hp-AFEM) in dimensions . The algorithm consists of iterating two routines: hp-NEARBEST finds a near-best -a…
Instance optimality of the adaptive maximum strategy
Lars Diening, Christian Kreuzer, Rob Stevenson
In this paper, we prove that the standard adaptive finite element method with a (modified) `maximum marking strategy' is `instance optimal' for the `total error', being the sum of…