Tristan Pryer

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.

Jan Giesselmann, Charalambos Makridakis, Tristan Pryer

We design consistent discontinuous Galerkin finite element schemes for the approximation of the Euler-Korteweg and the Navier-Stokes-Korteweg systems. We show that the scheme for the Euler-Korteweg system is energy and mass conservative and that the scheme for the Navier-Stokes-Korteweg system is mass conservative and monotonically energy dissipative. In this case the dissipation is isolated to viscous effects, that is, there is no numerical dissipation. In this sense the methods is consistent with the energy dissipation of the continuous PDE systems.

Emmanuil H. Georgoulis, Tristan Pryer

We introduce a family of Galerkin finite element methods which are constructed via recovery operators over element-wise discontinuous approximation spaces. This new family, termed collectively as recovered finite element methods (R-FEM) has a number of attractive features over both classical finite element and discontinuous Galerkin approaches, most important of which is its potential to produce stable conforming approximations in a variety of settings. Moreover, for special choices of recovery operators, R-FEM produces the same approximate solution as the classical conforming finite element method, while, trivially, one can recast (primal formulation) discontinuous Galerkin methods. A priori error bounds are shown for linear second order boundary value problems, verifying the optimality of the proposed method. Residual-type a posteriori bounds are also derived, highlighting the potential of R-FEM in the context of adaptive computations. Numerical experiments highlight the good approximation properties of the method in practice. A discussion on the potential use of R-FEM in various settings is also included.

Omar Lakkis, Tristan Pryer

We present a continuous finite element method for some examples of fully nonlinear elliptic equation. A key tool is the discretisation proposed in Lakkis & Pryer (2011, SISC) allowing us to work directly on the strong form of a linear PDE. An added benefit to making use of this discretisation method is that a recovered (finite element) Hessian is a biproduct of the solution process. We build on the linear basis and ultimately construct two different methodologies for the solution of second order fully nonlinear PDEs. Benchmark numerical results illustrate the convergence properties of the scheme for some test problems including the Monge-Ampère equation and Pucci's equation.

Omar Lakkis, Tristan Pryer

We propose a numerical method to approximate the solution of second order elliptic problems in nonvariational form. The method is of Galerkin type using conforming finite elements and applied directly to the nonvariational (nondivergence) form of a second order linear elliptic problem. The key tools are an appropriate concept of a 'finite element Hessian' and a Schur complement approach to solving the resulting linear algebra problem. The method is illustrated with computational experiments on three linear and one quasilinear PDE, all in nonvariational form.

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.

Emine Kesici, Beatrice Pelloni, Tristan Pryer et al.

We present the numerical solution of two-point boundary value problems for a third order linear PDE, representing a linear evolution in one space dimension. The difficulty of this problem is in the numerical imposition of the boundary conditions, and to our knowledge, no such computations exist. Instead of computing the evolution numerically, we evaluate the solution representation formula obtained by the unified transform of Fokas. This representation involves complex line integrals, but in order to evaluate these integrals numerically, it is necessary to deform the integration contours using appropriate deformation mappings. We formulate a strategy to implement effectively this deformation, which allows us to obtain accurate numerical results.

Nikos Katzourakis, Tristan Pryer

Given a map $u : Ω\subseteq \mathbb{R}^n \longrightarrow \mathbb{R}^N$, the $\infty$-Laplacian is the system \[ \label{1} Δ_\infty u \, :=\, \Big(\text{D}u \otimes \text{D}u + |\text{D}u|^2 [\text{D}u]^\bot \! \otimes I \Big) : \text{D}^2 u\, = \, 0. \tag{1} \] \eqref{1} is the model system of vectorial Calculus of Variations in $L^\infty$ and arises as the "Euler-Lagrange" equation in relation to the supremal functional \[ \label{2} E_\infty(u,Ω)\, :=\, \| \text{D}u \|_{L^\infty(Ω)}. \tag{2} \] The scalar case of \eqref{1} has been introduced by Aronsson in the 1960s and by now is relatively classical and well understood. The general system \eqref{1} has been discovered and studied by the first author in a series of recent papers. Supremal functionals are fundamental for applications because they provide more realistic models as opposed to conventional integral models. Herein we provide numerical approximations of solutions to the Dirichlet problem when $n=2$ and $N=2,3$ for certain carefully selected boundary data on the unit square. Our experiments demonstrate interesting and unexpected phenomena occurring in $L^\infty$ and provide insights on the structure of general solutions and the natural separation to phases they present.

Omar Lakkis, Charalambos Makridakis, Tristan Pryer

We use the elliptic reconstruction technique in combination with a duality approach to prove aposteriori error estimates for fully discrete back- ward Euler scheme for linear parabolic equations. As an application, we com- bine our result with the residual based estimators from the aposteriori esti- mation for elliptic problems to derive space-error indicators and thus a fully practical version of the estimators bounding the error in the L \infty (0, T ; L2(Ω)) norm. These estimators, which are of optimal order, extend those introduced by Eriksson and Johnson (1991) by taking into account the error induced by the mesh changes and allowing for a more flexible use of the elliptic estima- tors. For comparison with previous results we derive also an energy-based aposteriori estimate for the L \infty (0, T ; L2(Ω))-error which simplifies a previous one given in Lakkis and Makridakis (2006). We then compare both estimators (duality vs. energy) in practical situations and draw conclusions.

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.

Nikos Katzourakis, Tristan Pryer

In [KP16] (arXiv:1605.07880) the authors introduced a second-order variational problem in $L^{\infty}$. The associated equation, coined the $\infty$-Bilaplacian, is a \emph{third order} fully nonlinear PDE given by $Δ^2_\infty u\, := (Δu)^3 | D (Δu) |^2 = 0.$ In this work we build a numerical method aimed at quantifying the nature of solutions to this problem which we call $\infty$-Biharmonic functions. For fixed $p$ we design a mixed finite element scheme for the pre-limiting equation, the $p$-Bilaplacian $Δ^2_p u\, := Δ(| Δu |^{p-2} Δu) = 0.$ We prove convergence of the numerical solution to the weak solution of $Δ^2_p u = 0$ and show that we are able to pass to the limit $p\to\infty$. We perform various tests aimed at understanding the nature of solutions of $Δ^2_\infty u$ and in 1-$d$ we prove convergence of our discretisation to an appropriate weak solution concept of this problem, that of $\mathcal D$-solutions.

Tristan Pryer

We study discontinuous Galerkin approximations of the $p$--biharmonic equation from a variational perspective. We propose a discrete variational formulation of the problem based on a appropriate definition of a finite element Hessian and study convergence of the method (without rates) using a weak lower semicontinuity argument. We present numerical experiments aimed at testing the robustness of the method. We also note a superconvergence effect for some values of $p$.

Ellya Kawecki, Omar Lakkis, Tristan Pryer

We address the numerical solution via Galerkin type methods of the Monge-Ampère equation with transport boundary conditions arising in optimal mass transport, geometric optics and computational mesh or grid movement techniques. This fully nonlinear elliptic problem admits a linearisation via a Newton-Raphson iteration, which leads to an oblique derivative boundary value problem for elliptic equations in nondivergence form. We discretise these by employing the nonvariational finite element method, which lead to empirically observed optimal convergence rates, provided recovery techinques are used to approximate the gradient and the Hessian of the unknown functions. We provide extensive numerical testing to illustrate the strengths of our approach and the potential applications in optics and mesh movement.

Elizabeth Mansfield, Tristan Pryer

In this work we prove a weak Noether type theorem for a class of variational problems which include broken extremals. We then use this result to prove discrete Noether type conservation laws for certain classes of finite element discretisation of a model elliptic problem. In addition we study how well the finite element scheme satisfies the continuous conservation laws arising from the application of Noether's 1st Theorem (E. Noether 1918). We summarise extensive numerical tests, illustrating the conservativity of the discrete Noether law using the $p$--Laplacian as an example.

Nikos Katzourakis, Tristan Pryer

In this paper we initiate the study of $2$nd order variational problems in $L^\infty$, seeking to minimise the $L^\infty$ norm of a function of the hessian. We also derive and study the respective PDE arising as the analogue of the Euler-Lagrange equation. Given $\mathrm{H}\in C^1(\mathbb{R}^{n\times n}_s)$, for the functional \[ \label{1} \mathrm{E}_\infty(u,\mathcal{O})\, =\, \big\| \mathrm{H}\big(\mathrm{D}^2 u\big) \big\|_{L^\infty(\mathcal{O})}, \ \ \ u\in W^{2,\infty}(Ω),\ \mathcal{O}\subseteq Ω, \tag{1} \] the associated equation is the fully nonlinear 3rd order PDE \[ \label{2} \mathrm{A}^2_\infty u\, :=\,\big(\mathrm{H}_X\big(\mathrm{D}^2u\big)\big)^{\otimes 3}:\big(\mathrm{D}^3u\big)^{\otimes 2}\, =\,0. \tag{2} \] Special cases arise when $\mathrm{H}$ is the Euclidean length of either the full hessian or of the Laplacian, leading to the $\infty$-Polylaplacian and the $\infty$-Bilaplacian respectively. We establish several results for \eqref{1} and \eqref{2}, including existence of minimisers, of absolute minimisers and of "critical point" generalised solutions, proving also variational characterisations and uniqueness. We also construct explicit generalised solutions and perform numerical experiments.

Jan Giesselmann, Tristan Pryer

We give an a priori analysis of a semi-discrete discontinuous Galerkin scheme approximating solutions to a model of multiphase elastodynamics which involves an energy density depending not only on the strain but also the strain gradient. A key component in the analysis is the reduced relative entropy stability framework developed in [Gie14]. We prove optimal bounds for the strain in an appropriate norm and suboptimal bounds for the velocity.

James Jackaman, Georgios Papamikos, Tristan Pryer

We design a consistent Galerkin scheme for the approximation of the vectorial modified Korteweg-de Vries equation. We demonstrate that the scheme conserves energy up to machine precision. In this sense the method is consistent with the energy balance of the continuous system. This energy balance ensures there is no numerical dissipation allowing for extremely accurate long time simulations free from numerical artifacts. Various numerical experiments are shown demonstrating the asymptotic convergence of the method with respect to the discretisation parameters. Some simulations are also presented that correctly capture the unusual interactions between solitons in the vectorial setting.

Jan Giesselmann, Tristan Pryer

We give an a posteriori analysis of a semi-discrete discontinuous Galerkin scheme approximating solutions to a model of multiphase elastodynamics, which involves an energy density depending not only on the strain but also the strain gradient. A key component in the analysis is the reduced relative entropy stability framework developed in [Giesselmann 2014]. This framework allows energy type arguments to be applied to continuous functions. Since we advocate the use of discontinuous Galerkin methods we make use of two families of reconstructions, one set of discrete reconstructions [Makridakis and Nochetto 2006] and a set of elliptic reconstructions [Makridakis and Nochetto 2003] to apply the reduced relative entropy framework in this setting.

Nikos Katzourakis, Tristan Pryer

Let $Ω$ be an open set. We consider the supremal functional \[ \tag{1} \label{1} \ \ \ \ \ \ \mathrm{E}_\infty (u,\mathcal{O})\, :=\, \| \mathrm D u \|_{L^\infty( \mathcal{O} )}, \ \ \ \mathcal{O} \subseteq Ω\text{ open}, \] applied to locally Lipschitz mappings $u : \mathbb R^n \supseteq Ω\longrightarrow \mathbb R^N$, where $n,N\in \mathbb N$. This is the model functional of Calculus of Variations in $L^\infty$. The area is developing rapidly, but the vectorial case of $N\geq 2$ is still poorly understood. Due to the non-local nature of \eqref{1}, usual minimisers are not truly optimal. The concept of so-called absolute minimisers is the primary contender in the direction of variational concepts. However, these cannot be obtained by direct minimisation and the question of their existence under prescribed boundary data is open when $n,N\geq 2$. Herein we present numerical experiments based on a new method recently proposed by the first author in the papers [33, 35].

Ansar Calloo, Matthew Evans, François Madiot et al.

We present a computational study of diffusion synthetic acceleration (DSA) for the monoenergetic, isotropically scattering $S_N$ transport equations, discretised in space by a polytopic discontinuous Galerkin method. Using a discrete ordinates angular discretisation, we construct the DSA correction with an interior-penalty diffusion operator and compare a classical symmetric interior penalty (SIP) formulation with a modified interior penalty (MIP) variant, together with homogeneous Dirichlet and Marshak (Robin) diffusion boundary conditions imposed weakly in the DG framework. We quantify the observed convergence behaviour of the resulting source iteration across variations in optical thickness, scattering ratio, angular quadrature, mesh refinement, polynomial degree and mesh anisotropy on families of bounded Voronoi meshes. The results show that MIP-based DSA remains robust across the parameter ranges tested, whereas SIP-based DSA can lose robustness in the intermediate regime. In challenging optically thick, highly scattering settings, the observed convergence factors for the MIP-based schemes are typically below $0.6$.

Tristan Pryer

In this note we show that conforming Galerkin approximations for p-harmonic functions tend to infinity-harmonic functions in the limit p \to \infty and h \to 0, where h denotes the Galerkin discretisation parameter.

Veronika Chronholm, Tristan Pryer

We develop numerical schemes and sensitivity methods for stochastic models of proton transport that couple energy loss, range straggling and angular diffusion. For the energy equation we introduce a logarithmic Milstein scheme that guarantees positivity and achieves strong order one convergence. For the angular dynamics we construct a Lie-group integrator. The combined method maintains the natural geometric invariants of the system. We formulate dose deposition as a regularised path-dependent functional, obtaining a pathwise sensitivity estimator that is consistent and implementable. Numerical experiments confirm that the proposed schemes achieve the expected convergence rates and provide stable estimates of dose sensitivities.

Tristan Pryer

The goal of this work is to illustrate the application of the nonvariational finite element method to a specific Monge--Ampère type nonlinear partial differential equation. The equation we consider is that of prescribed Gauss curvature.

Aaron Pim, Tristan Pryer

We develop a multilevel Monte Carlo (MLMC) framework for uncertainty quantification with Monte Carlo dropout. Treating dropout masks as a source of epistemic randomness, we define a fidelity hierarchy by the number of stochastic forward passes used to estimate predictive moments. We construct coupled coarse--fine estimators by reusing dropout masks across fidelities, yielding telescoping MLMC estimators for both predictive means and predictive variances that remain unbiased for the corresponding dropout-induced quantities while reducing sampling variance at fixed evaluation budget. We derive explicit bias, variance and effective cost expressions, together with sample-allocation rules across levels. Numerical experiments on forward and inverse PINNs--Uzawa benchmarks confirm the predicted variance rates and demonstrate efficiency gains over single-level MC-dropout at matched cost.

Aaron Pim, Tristan Pryer

Accurate proton dose calculation using Monte Carlo (MC) is computationally demanding in workflows like robust optimisation, adaptive replanning, and probabilistic inference, which require repeated evaluations. To address this, we develop a neural surrogate that integrates Monte Carlo dropout to provide fast, differentiable dose predictions along with voxelwise predictive uncertainty. The method is validated through a series of experiments, starting with a one-dimensional analytic benchmark that establishes accuracy, convergence, and variance decomposition. Two-dimensional bone-water phantoms, generated using TOPAS Geant4, demonstrate the method's behavior under domain heterogeneity and beam uncertainty, while a three-dimensional water phantom confirms scalability for volumetric dose prediction. Across these settings, we separate epistemic (model) from parametric (input) contributions, showing that epistemic variance increases under distribution shift, while parametric variance dominates at material boundaries. The approach achieves significant speedups over MC while retaining uncertainty information, making it suitable for integration into robust planning, adaptive workflows, and uncertainty-aware optimisation in proton therapy.

Jan Giesselmann, Tristan Pryer

In this work we present an a posteriori error indicator for approximation schemes of Runge-Kutta-discontinuous-Galerkin type arising in applications of compressible fluid flows. The purpose of this indicator is not only for mesh adaptivity, we also make use of this to drive model adaptivity. This is where a perhaps costly complex model and a cheaper simple model are solved over different parts of the domain. The a posteriori bound we derive indicates the regions where the complex model can be relatively well approximated with the cheaper one. One such example which we choose to highlight is that of the Navier-Stokes-Fourier equations approximated by Euler's equations.

Tristan Pryer

In this work we present an a posteriori analysis for classes of inconsistent, nonconforming schemes approximating elliptic problems. We show the estimates coincide with existing ones for interior penalty type discontinuous Galerkin approximations of the Laplacian and give new estimates for inconsistent discontinuous Galerkin approximation schemes of elliptic problems under quadrature approximation. We also examine the effect of inconsistencies on the a posteriori analysis of schemes applied to an unbalanced problem.

Andreas Dedner, Tristan Pryer

We extend the finite element method introduced by Lakkis and Pryer [2011] to approximate the solution of second order elliptic problems in nonvariational form to incorporate the discontinuous Galerkin (DG) framework. This is done by viewing the NVFEM as a mixed method whereby the finite element Hessian is an auxiliary variable in the formulation. Representing the finite element Hessian in a discontinuous setting yields a linear system of the same size and having the same sparsity pattern of the compact DG methods for variational elliptic problems. Furthermore, the system matrix is very easy to assemble, Thus this approach greatly reduces the computational complexity of the discretisation compared to the continuous approach. We conduct a stability and consistency analysis making use of the unified framework set out in Arnold et. al. [2001]. We also give an apriori analysis of the method. The analysis applies to any consistent representation of the finite element Hessian, thus is applicable to the previous works making use of continuous Galerkin approximations.

Omar Lakkis, Tristan Pryer

We derive energy-norm aposteriori error bounds, using gradient recovery (ZZ) estimators to control the spatial error, for fully discrete schemes for the linear heat equation. This appears to be the first completely rigorous derivation of ZZ estimators for fully discrete schemes for evolution problems, without any restrictive assumption on the timestep size. An essential tool for the analysis is the elliptic reconstruction technique. Our theoretical results are backed with extensive numerical experimentation aimed at (a) testing the practical sharpness and asymptotic behaviour of the error estimator against the error, and (b) deriving an adaptive method based on our estimators. An extra novelty provided is an implementation of a coarsening error "preindicator", with a complete implementation guide in ALBERTA.