scholarly journals Discontinuous Galerkin time discretization methods for parabolic problems with linear constraints

2019 ◽  
Vol 27 (3) ◽  
pp. 155-182 ◽  
Author(s):  
Igor Voulis ◽  
Arnold Reusken
Keyword(s):  
Discontinuous Galerkin ◽  
Error Bounds ◽  
Convergence Rates ◽  
Time Discretization ◽  
Linear Constraints ◽  
Dirichlet Boundary Conditions ◽  
Time Dependent ◽  
Parabolic Problems ◽  
Discretization Methods ◽  
Optimal Convergence

Abstract We consider time discretization methods for abstract parabolic problems with inhomogeneous linear constraints. Prototype examples that fit into the general framework are the heat equation with inhomogeneous (time-dependent) Dirichlet boundary conditions and the time-dependent Stokes equation with an inhomogeneous divergence constraint. Two common ways of treating such linear constraints, namely explicit or implicit (via Lagrange multipliers) are studied. These different treatments lead to different variational formulations of the parabolic problem. For these formulations we introduce a modification of the standard discontinuous Galerkin (DG) time discretization method in which an appropriate projection is used in the discretization of the constraint. For these discretizations (optimal) error bounds, including superconvergence results, are derived. Discretization error bounds for the Lagrange multiplier are presented. Results of experiments confirm the theoretically predicted optimal convergence rates and show that without the modification the (standard) DG method has sub-optimal convergence behavior.

A Priori Error Analysis of a Discontinuous Galerkin Scheme for the Magnetic Induction Equation

2020 ◽  
Vol 20 (1) ◽  
pp. 121-140 ◽  
Author(s):  
Tanmay Sarkar
Keyword(s):  
Error Analysis ◽  
Discontinuous Galerkin ◽  
Magnetic Induction ◽  
Time Discretization ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Galerkin Scheme ◽  
Optimal Convergence ◽  
Discontinuous Galerkin Scheme ◽  
Runge Kutta

AbstractWe perform the error analysis of a stabilized discontinuous Galerkin scheme for the initial boundary value problem associated with the magnetic induction equations using standard discontinuous Lagrange basis functions. In order to obtain the quasi-optimal convergence incorporating second-order Runge–Kutta schemes for time discretization, we need a strengthened {4/3}-CFL condition ({\Delta t\sim h^{4/3}}). To overcome this unusual restriction on the CFL condition, we consider the explicit third-order Runge–Kutta scheme for time discretization. We demonstrate the error estimates in {L^{2}}-sense and obtain quasi-optimal convergence for smooth solution in space and time for piecewise polynomials with any degree {l\geq 1} under the standard CFL condition.

scholarly journals Space–time discontinuous Galerkin approximation of acoustic waves with point singularities

2020 ◽  
Author(s):  
Pratyuksh Bansal ◽  
Andrea Moiola ◽  
Ilaria Perugia ◽  
Christoph Schwab
Keyword(s):  
Wave Equation ◽  
Discontinuous Galerkin ◽  
Acoustic Waves ◽  
Convergence Rates ◽  
Galerkin Approximation ◽  
Space Time ◽  
Convergence Theory ◽  
Polygonal Domains ◽  
Optimal Convergence ◽  
Optimal Convergence Rates

Abstract We develop a convergence theory of space–time discretizations for the linear, second-order wave equation in polygonal domains $\varOmega \subset{\mathbb R}^2$, possibly occupied by piecewise homogeneous media with different propagation speeds. Building on an unconditionally stable space–time DG formulation developed in Moiola & Perugia (2018, A space–time Trefftz discontinuous Galerkin method for the acoustic wave equation in first-order formulation. Numer. Math., 138, 389–435), we (a) prove optimal convergence rates for the space–time scheme with local isotropic corner mesh refinement on the spatial domain, and (b) demonstrate numerically optimal convergence rates of a suitable sparse space–time version of the DG scheme. The latter scheme is based on the so-called combination formula, in conjunction with a family of anisotropic space–time DG discretizations. It results in optimal-order convergent schemes, also in domains with corners, with a number of degrees of freedom that scales essentially like the DG solution of one stationary elliptic problem in $\varOmega $ on the finest spatial grid. Numerical experiments for both smooth and singular solutions support convergence rate optimality on spatially refined meshes of the full and sparse space–time DG schemes.

scholarly journals A posteriori error bounds for fully-discrete hp-discontinuous Galerkin timestepping methods for parabolic problems

2021 ◽  
Author(s):  
Emmanuil H. Georgoulis ◽  
Omar Lakkis ◽  
Thomas P. Wihler
Keyword(s):  
Discontinuous Galerkin ◽  
Error Bounds ◽  
Posteriori Error ◽  
Parabolic Problems ◽  
Error Estimator ◽  
A Posteriori ◽  
A Posteriori Error ◽  
Fully Discrete ◽  
Time Space ◽  
A Posteriori Error Bounds

AbstractWe consider fully discrete time-space approximations of abstract linear parabolic partial differential equations (PDEs) consisting of an hp-version discontinuous Galerkin (DG) time stepping scheme in conjunction with standard (conforming) Galerkin discretizations in space. We derive abstract computable a posteriori error bounds resulting, for instance, in concrete bounds in "Equation missing"- and "Equation missing"-type norms when I is the temporal and "Equation missing" the spatial domain for the PDE. We base our methodology for the analysis on a novel space-time reconstruction approach. Our approach is flexible as it works for any type of elliptic error estimator and leaves their choice to the user. It also exhibits mesh-change estimators in a clear and concise way. We also show how our approach allows the derivation of such bounds in the "Equation missing" norm.

scholarly journals Optimally convergent hybridizable discontinuous Galerkin method for fifth-order Korteweg-de Vries type equations

2018 ◽  
Vol 52 (6) ◽  
pp. 2283-2306 ◽  
Author(s):  
Yanlai Chen ◽  
Bo Dong ◽  
Jiahua Jiang
Keyword(s):  
Discontinuous Galerkin ◽  
Convergence Rates ◽  
Time Derivative ◽  
Optimal Convergence ◽  
Hybridizable Discontinuous Galerkin ◽  
De Vries ◽  
Optimal Convergence Rates ◽  
Spatial Derivatives ◽  
Kdv Type Equations ◽  
Fifth Order

We develop and analyze the first hybridizable discontinuous Galerkin (HDG) method for solving fifth-order Korteweg-de Vries (KdV) type equations. We show that the semi-discrete scheme is stable with proper choices of the stabilization functions in the numerical traces. For the linearized fifth-order equations, we prove that the approximations to the exact solution and its four spatial derivatives as well as its time derivative all have optimal convergence rates. The numerical experiments, demonstrating optimal convergence rates for both the linear and nonlinear equations, validate our theoretical findings.

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