69 citations · 229 across the 40 of their papers we have counts for
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Multilevel Monte Carlo for Polytopic DG Discretisations with Spatially Represented Random Coefficients
Veronika Chronholm, Matt Evans, Emmanuil H. Georgoulis +1
We develop a multilevel Monte Carlo framework for coercive elliptic problems with random diffusion and reaction coefficients on polytopic meshes, with particular emphasis on the ap…
Hypocoercivity-preserving space-time Galerkin methods for kinetic Fokker-Planck equations
Zhaonan Dong, Emmanuil H. Georgoulis
We design and analyse a family of hypocoercivity-preserving fully discrete Galerkin methods for the (inhomogeneous) kinetic Fokker--Planck (kFP) equations, a class of evolution PDE…
-Version robust interior penalty discontinuous Galerkin methods for the -Laplacian on simplicial and on essentially arbitrarily-shaped element meshes
Emmanuil H. Georgoulis, Panagiotis Paraschis
We consider the discretization of the -Laplacian equation with an interior penalty discontinuous Galerkin method. We prove novel trace-type inverse estimates, leading to uncondi…
A nodally bound-preserving composite discontinuous Galerkin method on polytopic meshes
Abdolreza Amiri, Gabriel R. Barrenechea, Emmanuil H. Georgoulis +1
We introduce a nodally bound-preserving Galerkin method for second-order elliptic problems on general polygonal/polyhedral, henceforth collectively termed as \emph{polytopic}, mesh…
A posteriori error analysis and adaptivity of a space-time finite element method for the wave equation in second order formulation
Zhaonan Dong, Emmanuil H. Georgoulis, Lorenzo Mascotto +1
We establish rigorous \emph{a posteriori} error bounds for a space-time finite element method of arbitrary order discretising linear wave problems in second order formulation. The…
Parallel Nodal Interior-Penalty Discontinuous Galerkin Methods for the Subsonic Compressible Navier-Stokes Equations: Applications to Vortical Flows and VIV Problems
Spiros Zafeiris, Emmanuil H. Georgoulis, George Papadakis
We present a Discontinuous Galerkin (DG) solver for the compressible Navier-Stokes system, designed for applications of technological and industrial interest in the subsonic region…
Space-time finite element methods for nonlinear wave equations via elliptic regularisation
Lehel Banjai, Emmanuil H. Georgoulis, Brian Hennessy
We present and analyse a new conforming space-time Galerkin discretisation of a semi-linear wave equation, based on a variational formulation derived from De Giorgi's elliptic regu…
Asymptotic numerical hypocoercivity of the space-time discontinuous Galerkin method for the Kolmogorov equation
Zhaonan Dong, Emmanuil H. Georgoulis, Philip J. Herbert
We are concerned with discretisations of the classical Kolmogorov equation by a standard space-time discontinuous Galerkin method. {The} Kolmogorov equation serves as simple, yet r…
An posteriori error estimator for discontinuous Galerkin discretisations of convection-diffusion problems with application to Earth's mantle convection simulations
Tiffany Barry, Andrea Cangiani, Samuel P. Cox +1
We present new aposteriori error estimates for the interior penalty discontinuous Galerkin method applied to non-stationary convection-diffusion equations. The focus is on strongly…
Level-set shape optimization via polytopic discontinuous Galerkin methods
Raphael S. Fernandes, Emmanuil H. Georgoulis, Alberto Paganini
We introduce a new level-set shape optimization approach based on polytopic (i.e., polygonal in two and polyhedral in three spatial dimensions) discontinuous Galerkin methods. The…
A hypocoercivity-exploiting stabilised finite element method for Kolmogorov equation
Zhaonan Dong, Emmanuil H. Georgoulis, Philip J. Herbert
We propose a new stabilised finite element method for the classical Kolmogorov equation. The latter serves as a basic model problem for large classes of kinetic-type equations and,…
A Nodally Bound-Preserving Finite Element Method
Gabriel Barrenechea, Emmanuil Georgoulis, Tristan Pryer +1
This work proposes a nonlinear finite element method whose nodal values preserve bounds known for the exact solution. The discrete problem involves a nonlinear projection operator…
A posteriori error estimates for discontinuous Galerkin methods on polygonal and polyhedral meshes
Andrea Cangiani, Zhaonan Dong, Emmanuil H. Georgoulis
We present a new residual-type energy-norm a posteriori error analysis for interior penalty discontinuous Galerkin (dG) methods for linear elliptic problems. The new error bounds a…
Discrete Gradient Flow Approximations of High Dimensional Evolution Partial Differential Equations via Deep Neural Networks
Emmanuil H. Georgoulis, Michail Loulakis, Asterios Tsiourvas
We consider the approximation of initial/boundary value problems involving, possibly high-dimensional, dissipative evolution partial differential equations (PDEs) using a deep neur…
Robust interior penalty discontinuous Galerkin methods
Zhaonan Dong, Emmanuil H. Georgoulis
Classical interior penalty discontinuous Galerkin (IPDG) methods for diffusion problems require a number of assumptions on the local variation of mesh-size, polynomial degree, and…
GPU-accelerated discontinuous Galerkin methods on polytopic meshes
Zhaonan Dong, Emmanuil H. Georgoulis, Thomas Kappas
Discontinuous Galerkin (dG) methods on meshes consisting of polygonal/polyhedral (henceforth, collectively termed as \emph{polytopic}) elements have received considerable attention…
Adaptive non-hierarchical Galerkin methods for parabolic problems with application to moving mesh and virtual element methods
Andrea Cangiani, Emmanuil H. Georgoulis, Oliver J. Sutton
We present a posteriori error estimates for inconsistent and non-hierarchical Galerkin methods for linear parabolic problems, allowing them to be used in conjunction with very gene…
Convergence of adaptive discontinuous Galerkin methods (corrected version of [Math. Comp. 87 (2018), no. 314, 2611--2640])
Christian Kreuzer, Emmanuil H. Georgoulis
We develop a general convergence theory for adaptive discontinuous Galerkin methods for elliptic PDEs covering the popular SIPG, NIPG and LDG schemes as well as all practically rel…
A posteriori error estimates for the Allen-Cahn problem
Konstantinos Chrysafinos, Emmanuil H. Georgoulis, Dimitra Plaka
This work is concerned with the proof of \emph{a posteriori} error estimates for fully-discrete Galerkin approximations of the Allen-Cahn equation in two and three spatial dimensio…
-Version discontinuous Galerkin methods on essentially arbitrarily-shaped elements
Andrea Cangiani, Zhaonan Dong, Emmanuil H. Georgoulis
We extend the applicability of the popular interior-penalty discontinuous Galerkin (dG) method discretizing advection-diffusion-reaction problems to meshes comprising extremely gen…
Hypocoercivity-compatible finite element methods for the long-time computation of Kolmogorov's equation
Emmanuil H. Georgoulis
This work is concerned with the development of a family of Galerkin finite element methods for the classical Kolmogorov's equation. Kolmogorov's equation serves as a sufficiently r…
Recovered finite element methods on polygonal and polyhedral meshes
Zhaonan Dong, Emmanuil H. Georgoulis, Tristan Pryer
Recovered finite element methods (R-FEM) have been recently introduced for meshes consisting of simplicial and/or box-type meshes. Here, utilising the flexibility of R-FEM framewor…
A posteriori error bounds for fully-discrete hp-discontinuous Galerkin timestepping methods for parabolic problems
Emmanuil H. Georgoulis, Omar Lakkis, Thomas P. Wihler
We consider fully discrete time-space approximations of abstract linear parabolic partial differential equations (PDEs) consisting of an -version discontinuous Galerkin (DG) ti…
Virtual Element Method for Quasilinear Elliptic Problems
Andrea Cangiani, Panagiotis Chatzipantelidis, Ganesh Diwan +1
We present a Virtual Element Method (VEM) for the solution of Dirichlet problems for the quasilinear equation with essential boundary conditions…