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This work is focused on the application of functional-type a posteriori error estimates and corresponding indicators to a class of time-dependent problems. We consider the algorithmic part of their derivation and implementation and also…
A general framework for the numerical approximation of evolution problems is presented that allows to preserve exactly an underlying Hamiltonian- or gradient structure. The approach relies on rewriting the evolution problem in a particular…
This work develops an energy-based discontinuous Galerkin (EDG) method for the nonlinear Schr\"odinger equation with the wave operator. The focus of the study is on the energy-conserving or energy-dissipating behavior of the method with…
This chapter reviews and compares discontinuous Galerkin time-stepping methods for the numerical approximation of second-order ordinary differential equations, particularly those stemming from space finite element discretization of wave…
In this study, we examine numerical approximations for 2nd-order linear-nonlinear differential equations with diverse boundary conditions, followed by the residual corrections of the first approximations. We first obtain numerical results…
The numerical simulation of complex physical processes requires the use of economical discrete models. This lecture presents a general paradigm of deriving a posteriori error estimates for the Galerkin finite element approximation of…
In this work, we consider space-time goal-oriented a posteriori error estimation for parabolic problems. Temporal and spatial discretizations are based on Galerkin finite elements of continuous and discontinuous type. The main objectives…
In this paper, we study deep neural networks (DNNs) for solving high-dimensional evolution equations with oscillatory solutions. Different from deep least-squares methods that deal with time and space variables simultaneously, we propose a…
We propose a numerical method for approximate calculations of the time evolution of point particle systems given only the system's Hamiltonian function and initial conditions. The method both generates and solves the equations of motion…
We consider variational time integration using continuous Galerkin Petrov methods applied to evolutionary systems of changing type. We prove optimal-order convergence of the error in a cGP-like norm and conclude the paper with some…
Multi-adaptive Galerkin methods are extensions of the standard continuous and discontinuous Galerkin methods for the numerical solution of initial value problems for ordinary or partial differential equations. In particular, the…
In this paper, we propose an adaptive approach, based on mesh refinement or parametric enrichment with polynomial degree adaption, for numerical solution of convection dominated equations with random input data. A parametric system emerged…
The nonlocality of the fractional operator causes numerical difficulties for long time computation of the time-fractional evolution equations. This paper develops a high-order fast time-stepping discontinuous Galerkin finite element method…
In this paper we develop an adaptive procedure for the numerical solution of semilinear parabolic problems, with possible singular perturbations. Our approach combines a linearization technique using Newton's method with an adaptive…
We present a new $hp$-version space-time discontinuous Galerkin (dG) finite element method for the numerical approximation of parabolic evolution equations on general spatial meshes consisting of polygonal/polyhedral (polytopic) elements,…
In this article we present an a posteriori error estimator for the spatial-stochastic error of a Galerkin-type discretisation of an initial value problem for a random hyperbolic conservation law. For the stochastic discretisation we use the…
We propose and analyze an iterative high-order hybridized discontinuous Galerkin (iHDG) discretization for linear partial differential equations. We improve our previous work (SIAM J. Sci. Comput. Vol. 39, No. 5, pp. S782--S808) in several…
Discontinuous Galerkin (DG) methods for the numerical solution of partial differential equations have enjoyed considerable success because they are both flexible and robust: They allow arbitrary unstructured geometries and easy control of…
The aim of the paper is to demonstrate the use of the Galerkin method for some kind of Volterra equations, determininistic and stochastic as well. The paper consists of two parts: the theoretical and numerical one. In the first part we…
In this paper we study the numerical method for approximating the random periodic solution of semiliear stochastic evolution equations. The main challenge lies in proving a convergence over an infinite time horizon while simulating…