Emmanuil H. Georgoulis

Andrea Cangiani, Emmanuil H. Georgoulis, Tristan Pryer et al.

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. Mesh adaptation is particularly simple to implement since elements with consecutive co-planar edges/faces are allowed and, therefore, locally adapted meshes do not require any local mesh post-processing.

Daniel Elfverson, Emmanuil H. Georgoulis, Axel Målqvist et al.

A convergence result for a discontinuous Galerkin multiscale method for a second order elliptic problem is presented. We consider a heterogeneous and highly varying diffusion coefficient in $L^\infty(Ω,\mathbb{R}^{d\times d}_{sym})$ with uniform spectral bounds and without any assumption on scale separation or periodicity. The multiscale method uses a corrected basis that is computed on patches/subdomains. The error, due to truncation of corrected basis, decreases exponentially with the size of the patches. Hence, to achieve an algebraic convergence rate of the multiscale solution on a uniform mesh with mesh size $H$ to a reference solution, it is sufficient to choose the patch sizes as $\mathcal{O}(H|\log(H^{-1})|)$. We also discuss a way to further localize the corrected basis to element-wise support leading to a slight increase of the dimension of the space. Improved convergence rate can be achieved depending on the piecewise regularity of the forcing function. Linear convergence in energy norm and quadratic convergence in $L^2$-norm is obtained independently of the forcing function. A series of numerical experiments confirms the theoretical rates of convergence.

Emmanuil H. Georgoulis, Jeremy Levesley, Fazli Subhan

A multilevel kernel-based interpolation method, suitable for moderately high-dimensional function interpolation problems, is proposed. The method, termed multilevel sparse kernel-based interpolation (MLSKI, for short), uses both level-wise and direction-wise multilevel decomposition of structured (or mildly unstructured) interpolation data sites in conjunction with the application of kernel-based interpolants with different scaling in each direction. The multilevel interpolation algorithm is based on a hierarchical decomposition of the data sites, whereby at each level the detail is added to the interpolant by interpolating the resulting residual of the previous level. On each level, anisotropic radial basis functions are used for solving a number of small interpolation problems, which are subsequently linearly combined to produce the interpolant. MLSKI can be viewed as an extension of $d$-boolean interpolation (which is closely related to ideas in sparse grid and hyperbolic crosses literature) to kernel-based functions, within the hierarchical multilevel framework to achieve accelerated convergence. Numerical experiments suggest that the new algorithm is numerically stable and efficient for the reconstruction of large data in $\mathbb{R}^{d}\times \mathbb{R}$, for $d = 2, 3, 4$, with tens or even hundreds of thousands data points. Also, MLSKI appears to be generally superior over classical radial basis function methods in terms of complexity, run time and convergence at least for large data sets.

Andrea Cangiani, Panagiotis Chatzipantelidis, Ganesh Diwan et al.

We present a Virtual Element Method (VEM) for the solution of Dirichlet problems for the quasilinear equation $-\text{div} (k(u)\text{grad} u)=f$ with essential boundary conditions. Within the VEM the nonlinear coefficient is evaluated with the piecewise polynomial projection of the virtual element ansatz. Well posedness of the discrete problem and optimal order a priori error estimates in the $H^1$ and $L^2$ norms are proven. In addition, the convergence of fixed point iterations for the solution of the resulting nonlinear system is established. Numerical examples confirm the convergence analysis.

Lehel Banjai, Emmanuil H. Georgoulis, Oluwaseun Lijoka

A new space-time discontinuous Galerkin (dG) method utilising special Trefftz polynomial basis functions is proposed and fully analysed for the scalar wave equation in a second order formulation. The dG method considered is motivated by the class of interior penalty dG methods, as well as by the classical work of Hughes and Hulbert. The choice of the penalty terms included in the bilinear form is essential for both the theoretical analysis and for the practical behaviour of the method for the case of lowest order basis functions. A best approximation result is proven for this new space-time dG method with Trefftz-type basis functions. Rates of convergence are proved in any dimension and verified numerically in spatial dimensions $d = 1$ and $d = 2$. Numerical experiments highlight the effectiveness of the Trefftz method in problems with energy at high frequencies.

Andrea Cangiani, Emmanuil H. Georgoulis, Irene Kyza et al.

This work is concerned with the development of a space-time adaptive numerical method, based on a rigorous a posteriori error bound, for a semilinear convection-diffusion problem which may exhibit blow-up in finite time. More specifically, a posteriori error bounds are derived in the $L^{\infty}(L^2)+L^2(H^1)$-type norm for a first order in time implicit-explicit (IMEX) interior penalty discontinuous Galerkin (dG) in space discretization of the problem, although the theory presented is directly applicable to the case of conforming finite element approximations in space. The choice of the discretization in time is made based on a careful analysis of adaptive time stepping methods for ODEs that exhibit finite time blow-up. The new adaptive algorithm is shown to accurately estimate the blow-up time of a number of problems, including one which exhibits regional blow-up.

Andrea Cangiani, John Chapman, Emmanuil Georgoulis et al.

We consider a finite element method which couples the continuous Galerkin method away from internal and boundary layers with a discontinuous Galerkin method in the vicinity of layers. We prove that this consistent method is stable in the streamline diffusion norm if the convection field flows non-characteristically from the region of the continuous Galerkin to the region of the discontinuous Galerkin method. The stability properties of the coupled method are illustrated by numerical experiments.

Emmanuil H. Georgoulis, Juha M. Virtanen

An adaptive algorithm, based on residual type a posteriori indicators of errors measured in $L^{\infty}(L^2)$ and $L^2(L^2)$ norms, for a numerical scheme consisting of implicit Euler method in time and discontinuous Galerkin method in space for linear parabolic fourth order problems is presented. The a posteriori analysis is performed for convex domains in two and three space dimensions for local spatial polynomial degrees $r\ge 2$. The a posteriori estimates are then used within an adaptive algorithm, highlighting their relevance in practical computations, which results into substantial reduction of computational effort.

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 framework, we extend their definition on polygonal and polyhedral meshes in two and three spatial dimensions, respectively. A key attractive feature of this framework is its ability to produce conforming discretizations, yet involving only as many degrees of freedom as discontinuous Galerkin methods over general polygonal/polyhedral meshes with potentially many faces per element. A priori error bounds are shown for general linear, possibly degenerate, second order advection-diffusion-reaction boundary value problems. A series of numerical experiments highlights the good practical performance of the proposed numerical framework.

Emmanuil Georgoulis, Charalambos Makridakis, Tristan Pryer

We prove the inf-sup stability of the interior penalty class of discontinuous Galerkin schemes in unbalanced mesh-dependent norms, under a mesh condition allowing for a general class of meshes, which includes many examples of geometrically graded element neighbourhoods. The inf-sup condition results in the stability of the interior penalty Ritz projection in $L^2$ as well as, for the first time, quasi-best approximations in the $L^2$-norm which in turn imply a priori error estimates that do not depend on the global maximum meshsize in that norm. Some numerical experiments are also given.

Emmanuil H. Georgoulis, Tristan Pryer

We prove the inf-sup stability of a discontinuous Galerkin scheme for second order elliptic operators in (unbalanced) mesh-dependent norms for quasi-uniform meshes for all spatial dimensions. This results in a priori error bounds in these norms. As an application we examine a problem with rough source term where the solution can not be characterised as a weak solution and show quasi-optimal error control.

Zhaonan Dong, Emmanuil H. Georgoulis, Jeremy Levesley et al.

A fast multilevel algorithm based on directionally scaled tensor-product Gaussian kernels on structured sparse grids is proposed for interpolation of high-dimensional functions and for the numerical integration of high-dimensional integrals. The algorithm is based on the recent Multilevel Sparse Kernel-based Interpolation (MLSKI) method (Georgoulis, Levesley \& Subhan, \emph{SIAM J. Sci. Comput.}, 35(2), pp.~A815--A831, 2013), with particular focus on the fast implementation of Gaussian-based MLSKI for interpolation and integration problems of high-dimen-sional functions $f:[0,1]^d\to\mathbb{R}$, with $5\le d\le 10$. The MLSKI interpolation procedure is shown to be interpolatory and a fast implementation is proposed. More specifically, exploiting the tensor-product nature of anisotropic Gaussian kernels, one-dimensional cardinal basis functions on a sequence of hierarchical equidistant nodes are precomputed to machine precision, rendering the interpolation problem into a fully parallelisable ensemble of linear combinations of function evaluations. A numerical integration algorithm is also proposed, based on interpolating the (high-dimensional) integrand. A series of numerical experiments highlights the applicability of the proposed algorithm for interpolation and integration for up to 10-dimensional problems.

Andrea Cangiani, Emmanuil H. Georgoulis, Max Jensen

A discontinuous Galerkin (dG) method for the numerical solution of initial/boundary value multi-compartment partial differential equation (PDE) models, interconnected with interface conditions, is presented and analysed. The study of interface problems is motivated by models of mass transfer of solutes through semi-permeable membranes. More specifically, a model problem consisting of a system of semilinear parabolic advection-diffusion-reaction partial differential equations in each compartment, equipped with respective initial and boundary conditions, is considered. Nonlinear interface conditions modelling selective permeability, congestion and partial reflection are applied to the compartment interfaces. An interior penalty dG method is presented for this problem and it is analysed in the space-discrete setting. The a priori analysis shows that the method yields optimal a priori bounds, provided the exact solution is sufficiently smooth. Numerical experiments indicate agreement with the theoretical bounds and highlight the stability of the numerical method in the advection-dominated regime.

Zhaonan Dong, Emmanuil H. Georgoulis, Lorenzo Mascotto et al.

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 method combines standard finite elements in space and continuous piecewise polynomials in time with an upwind discontinuous Galerkin-type approximation for the second temporal derivative. The proposed scheme accepts dynamic mesh modification, as required by space-time adaptive algorithms, resulting in a discontinuous temporal discretisation when mesh changes occur. We prove \emph{a posteriori} error bounds in the $L^\infty(L^2)$-norm, using carefully designed temporal and spatial reconstructions; explicit control on the constants (including the spatial and temporal orders of the method) in those error bounds is shown. The convergence behaviour of an error estimator is verified numerically, also taking into account the effect of the mesh change. A space-time adaptive algorithm is proposed and tested numerically.

Andrea Cangiani, Emmanuil H. Georgoulis, Stephen Metcalfe

This work is concerned with the derivation of a robust a posteriori error estimator for a discontinuous Galerkin method discretisation of linear non-stationary convection-diffusion initial/boundary value problems and with the implementation of a corresponding adaptive algorithm. More specifically, we derive a posteriori bounds for the error in the $L^2(H^1)$-type norm for an interior penalty discontinuous Galerkin (dG) discretisation in space and a backward Euler discretisation in time. An important feature of the estimator is robustness with respect to the Péclet number of the problem which is verified in practice by a series of numerical experiments. Finally, an adaptive algorithm is proposed utilising the error estimator. Optimal rate of convergence of the adaptive algorithm is observed in a number of test problems.

Emmanuil H. Georgoulis, Panagiotis Paraschis

We consider the discretization of the $p$-Laplacian equation with an interior penalty discontinuous Galerkin method. We prove novel trace-type inverse estimates, leading to unconditional stability of the method. Further, $hp$-version a priori norm and quasi-norm error estimates are established, subordinate to available polynomial approximation results. The analysis is extended to discontinuous Galerkin methods, based on meshes with essentially arbitrarily-shaped, curved polygonal/polyhedral elements. This extension requires the proof of new $hp$-version weighted inverse estimates on essentially arbitrarily-shaped elements. Numerical experiments are also presented, highlighting the relevance of the theoretical findings.

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 PDEs with degenerate diffusion. The proposed methods mimic Villani's framework of enhanced quadratic forms [23], yielding a coercive bilinear form in an exponentially weighted norm that admits a spectral gap/Poincaré inequality despite the degeneracy. The problem is formulated as a fourth-order-in-space evolution PDE on the whole space $\mathbb{R}^{d}\times\mathbb{R}^d$. The spatial discretisation employs continuous piecewise polynomial finite element spaces on simplicial and/or box-type meshes comprising both finite and ``infinite'' elements, while nonconformity is handled by numerical fluxes in the spirit of $C^0$ interior penalty ($C^0$-IP) methods. The analysis requires new polynomial inverse trace inequalities in exponentially weighted norms for simplicial, box-type, and semi-infinite prismatic elements, which are proved for a broad class of exponential weights and are of independent interest. Coercivity of the Galerkin method then leads to exponential convergence to equilibrium via an exponentially weighted Poincaré inequality. We further develop a fully discrete scheme by coupling the spatial discretisation with an $hp$-version discontinuous Galerkin time-stepping method of arbitrary order and establish the same exponential convergence. The proposed methods preserve the total mass and exhibit \emph{provably} exponential convergence to equilibrium, making them well suited for long-time kFP simulations. Numerical experiments validate the theoretical results and demonstrate the convergence behaviour of the proposed methods.

Emmanuil H. Georgoulis, Antonis Papapantoleon, Costas Smaragdakis

We develop a novel deep learning approach for pricing European basket options written on assets that follow jump-diffusion dynamics. The option pricing problem is formulated as a partial integro-differential equation, which is approximated via a new implicit-explicit minimizing movement time-stepping approach, involving approximation by deep, residual-type Artificial Neural Networks (ANNs) for each time step. The integral operator is discretized via two different approaches: (a) a sparse-grid Gauss-Hermite approximation following localised coordinate axes arising from singular value decompositions, and (b) an ANN-based high-dimensional special-purpose quadrature rule. Crucially, the proposed ANN is constructed to ensure the appropriate asymptotic behavior of the solution for large values of the underlyings and also leads to consistent outputs with respect to a priori known qualitative properties of the solution. The performance and robustness with respect to the dimension of these methods are assessed in a series of numerical experiments involving the Merton jump-diffusion model, while a comparison with the deep Galerkin method and the deep BSDE solver with jumps further supports the merits of the proposed approach.