Corpus ID: 229371400

A Space-Time DPG Method for the Heat Equation

  title={A Space-Time DPG Method for the Heat Equation},
  author={L. Diening and J. Storn},
This paper introduces an ultra-weak space-time DPG method for the heat equation. We prove well-posedness of the variational formulation with broken test functions and verify quasi-optimality of a practical DPG scheme. Numerical experiments visualize beneficial properties of an adaptive and parabolically scaled mesh-refinement driven by the built-in error control of the DPG method. 
4 Citations

Figures and Tables from this paper

Analysis of Backward Euler Primal DPG Methods
Optimal error estimates are shown in a natural norm and in the L2{L^{2}} norm of the field variable for the heat equation of a primal DPG formulation of a class of parabolic problems. Expand
A wavelet-in-time, finite element-in-space adaptive method for parabolic evolution equations
In this work, an r-linearly converging adaptive solver is constructed for parabolic evolution equations in a simultaneous space-time variational formulation. Exploiting the product structure of theExpand
Minimal residual space-time discretizations of parabolic equations: Asymmetric spatial operators
A minimal residual discretization of a simultaneous spacetime variational formulation of parabolic evolution equations is considered and quasi-optimality of the numerical approximations without assuming symmetry of the spatial part of the differential operator is shown. Expand
Optimal local approximation spaces for parabolic problems
Local space-time approximation spaces for parabolic problems that are optimal in the sense of Kolmogorov and may be employed in multiscale and domain decomposition methods are proposed and derive rigorous local and global a priori error bounds. Expand


Low-order dPG-FEM for an elliptic PDE
This paper introduces a novel lowest-order discontinuous PetrovGalerkin (dPG) finite element method (FEM) for the Poisson model problem that allows for a direct proof of the discrete infsup condition and a complete apriori and aposteriori error analysis. Expand
A Class of Discontinuous Petrov–Galerkin Methods. Part I: The Transport Equation
Considering a simple model transport problem, we present a new finite element method. While the new method fits in the class of discontinuous Galerkin (DG) methods, it differs from standard DG andExpand
Stability of Galerkin discretizations of a mixed space–time variational formulation of parabolic evolution equations
Galerkin discretizations of a new well-posed mixed space-time variational formulation of parabolic PDEs are analyzed and the resulting Galerkin operators are shown to be uniformly stable. Expand
Space-time least-squares finite elements for parabolic equations
This work presents a space-time least squares finite element method for the heat equation based on residual minimization in L2 norms in space- time of an equivalent first order system and presents a-priori error analysis on simplicialspace-time meshes which are highly structured. Expand
On the stability of DPG formulations of transport equations
It is shown under mild assumptions on the convection field that piecewise polynomial test spaces of degree m+1 over a refinement of the primal partition with uniformly bounded refinement depth give rise to uniformly stable Petrov-Galerkin discretizations. Expand
A Spacetime DPG Method for the Schrödinger Equation
A spacetime discontinuous Petrov--Galerkin (DPG) method for the linear time-dependent Schrodinger equation is proposed and two variational formulations are proved to be well posed. Expand
Dispersive and Dissipative Errors in the DPG Method with Scaled Norms for Helmholtz Equation
This paper studies the discontinuous Petrov--Galerkin (DPG) method, where the test space is normed by a modified graph norm. The modification scales one of the terms in the graph norm by an arbitraryExpand
An analysis of the practical DPG method
A complete error analysis of the Discontinuous Petrov Galerkin (DPG) method is given, accounting for all the approximations made in its practical implementation, and it is shown that this approximation maintains optimal convergence rates. Expand
An ultraweak formulation of the Kirchhoff-Love plate bending model and DPG approximation
An ultraweak variational formulation for a variant of the Kirchhoff-Love plate bending model is developed and a discretization of the discontinuous Petrov-Galerkin type with optimal test functions (DPG) is introduced, proving well-posedness of the ultraweak formulation and quasi-optimal convergence of the DPG scheme. Expand
Breaking spaces and forms for the DPG method and applications including Maxwell equations
The equivalence of stability for various formulations of the same Maxwell problem is proved, including the strong form, the ultraweak form, and various forms in between. Expand