An official website of the United States government
Abstract This paper is about learning the parameter-to-solution map for systems of partial differential equations (PDEs) that depend on a potentially large number of parameters covering all PDE types for which a stable variational formulation (SVF) can be found. A central constituent is the notion of variationally correct residual loss function, meaning that its value is always uniformly proportional to the squared solution error in the norm determined by the SVF, hence facilitating rigorous a posteriori accuracy control. It is based on a single variational problem, associated with the family of parameter-dependent fibre problems, employing the notion of direct integrals of Hilbert spaces. Since in its original form the loss function is given as a dual test norm of the residual; a central objective is to develop equivalent computable expressions. The first critical role is played by hybrid hypothesis classes, whose elements are piecewise polynomial in (low-dimensional) spatio-temporal variables with parameter-dependent coefficients that can be represented, for example, by neural networks. Second, working with first-order SVFs we distinguish two scenarios: (i) the test space can be chosen as an $$L_{2}$$-space (such as for elliptic or parabolic problems) so that residuals can be evaluated directly as elements of $$L_{2}$$; (ii) when trial and test spaces for the fibre problems depend on the parameters (as for transport equations) we use ultra-weak formulations. In combination with discontinuous Petrov–Galerkin concepts the hybrid format is then instrumental to arrive at variationally correct computable residual loss functions. Our findings are illustrated by numerical experiments representing (i) and (ii), namely elliptic boundary value problems with piecewise constant diffusion coefficients and pure transport equations with parameter-dependent convection fields.
more »
« less
Adaptive Strategies for Transport Equations
Abstract This paper is concerned with a posteriori error bounds for linear transport equations and related questions of contriving corresponding adaptive solution strategies in the context of Discontinuous Petrov Galerkin schemes. After indicating our motivation for this investigation in a wider context the first major part of the paper is devoted to the derivation and analysis of a posteriori error bounds that, under mild conditions on variable convection fields, are efficient and, modulo a data-oscillation term, reliable. In particular, it is shown that these error estimators are computed at a cost that stays uniformly proportional to the problem size. The remaining part of the paper is then concerned with the question whether typical bulk criteria known from adaptive strategies for elliptic problems entail a fixed error reduction rate also in the context of transport equations. This turns out to be significantly more difficult than for elliptic problems and at this point we can give a complete affirmative answer for a single spatial dimension. For the general multidimensional case we provide partial results which we find of interest in their own right. An essential distinction from known concepts is that global arguments enter the issue of error reduction. An important ingredient of the underlying analysis, which is perhaps interesting in its own right, is to relate the derived error indicators to the residuals that naturally arise in related least squares formulations. This reveals a close interrelation between both settings regarding error reduction in the context of adaptive refinements.
more »
« less
- Award ID(s):
- 1720297
- PAR ID:
- 10157969
- Date Published:
- Journal Name:
- Computational Methods in Applied Mathematics
- Volume:
- 19
- Issue:
- 3
- ISSN:
- 1609-4840
- Page Range / eLocation ID:
- 431 to 464
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
More Like this
-
-
Abstract We derive a residual-based a posteriori error estimator for the conforming hp -Adaptive Finite Element Method ( hp -AFEM) for the steady state Stokes problem describing the slow motion of an incompressible fluid. This error estimator is obtained by extending the idea of a posteriori error estimation for the classical h -version of AFEM. We also establish the reliability and efficiency of the error estimator. The proofs are based on the well-known Clément-type interpolation operator introduced in [28] in the context of the hp -AFEM. Numerical experiments show the performance of an adaptive hp-FEM algorithm using the proposed a posteriori error estimator.more » « less
-
We are concerned with free boundary problems arising from the analysis of multidimensional transonic shock waves for the Euler equations in compressible fluid dynamics. In this expository paper, we survey some recent developments in the analysis of multidimensional transonic shock waves and corresponding free boundary problems for the compressible Euler equations and related nonlinear partial differential equations (PDEs) of mixed type. The nonlinear PDEs under our analysis include the steady Euler equations for potential flow, the steady full Euler equations, the unsteady Euler equations for potential flow, and related nonlinear PDEs of mixed elliptic–hyperbolic type. The transonic shock problems include the problem of steady transonic flow past solid wedges, the von Neumann problem for shock reflection–diffraction, and the Prandtl–Meyer problem for unsteady supersonic flow onto solid wedges. We first show how these longstanding multidimensional transonic shock problems can be formulated as free boundary problems for the compressible Euler equations and related nonlinear PDEs of mixed type. Then we present an effective nonlinear method and related ideas and techniques to solve these free boundary problems. The method, ideas, and techniques should be useful to analyze other longstanding and newly emerging free boundary problems for nonlinear PDEs.more » « less
-
We prove that the time of classical existence of smooth solutions to the relativistic Euler equations can be bounded from below in terms of norms that measure the “(sound) wave-part” of the data in Sobolev space and “transport-part” in higher regularity Sobolev space and Hölder spaces. The solutions are allowed to have nontrivial vorticity and entropy. We use the geometric framework from [M. M. Disconzi and J. Speck, The relativistic Euler equations: Remarkable null structures and regularity properties, Ann. Henri Poincaré 20(7) (2019) 2173–2270], where the relativistic Euler flow is decomposed into a “wave-part”, that is, geometric wave equations for the velocity components, density and enthalpy, and a “transport-part”, that is, transport-div-curl systems for the vorticity and entropy gradient. Our main result is that the Sobolev norm [Formula: see text] of the variables in the “wave-part” and the Hölder norm [Formula: see text] of the variables in the “transport-part” can be controlled in terms of initial data for short times. We note that the Sobolev norm assumption [Formula: see text] is the optimal result for the variables in the “wave-part”. Compared to low-regularity results for quasilinear wave equations and the three-dimensional (3D) non-relativistic compressible Euler equations, the main new challenge of the paper is that when controlling the acoustic geometry and bounding the wave equation energies, we must deal with the difficulty that the vorticity and entropy gradient are four-dimensional space-time vectors satisfying a space-time div-curl-transport system, where the space-time div-curl part is not elliptic. Due to lack of ellipticity, one cannot immediately rely on the approach taken in [M. M. Disconzi and J. Speck, The relativistic Euler equations: Remarkable null structures and regularity properties, Ann. Henri Poincaré 20(7) (2019) 2173–2270] to control these terms. To overcome this difficulty, we show that the space-time div-curl systems imply elliptic div-curl-transport systems on constant-time hypersurfaces plus error terms that involve favorable differentiations and contractions with respect to the four-velocity. By using these structures, we are able to adequately control the vorticity and entropy gradient with the help of energy estimates for transport equations, elliptic estimates, Schauder estimates and Littlewood–Paley theory.more » « less
-
The phase field method is becoming the de facto choice for the numerical analysis of complex problems that involve multiple initiating, propagating, interacting, branching and merging fractures. However, within the context of finite element modelling, the method requires a fine mesh in regions where fractures will propagate, in order to capture sharp variations in the phase field representing the fractured/damaged regions. This means that the method can become computationally expensive when the fracture propagation paths are not known a priori. This paper presents a 2D hp-adaptive discontinuous Galerkin finite element method for phase field fracture that includes a posteriori error estimators for both the elasticity and phase field equations, which drive mesh adaptivity for static and propagating fractures. This combination means that it is possible to be reliably and efficiently solve phase field fracture problems with arbitrary initial meshes, irrespective of the initial geometry or loading conditions. This ability is demonstrated on several example problems, which are solved using a light-BFGS (Broyden–Fletcher–Goldfarb–Shanno) quasi-Newton algorithm. The examples highlight the importance of driving mesh adaptivity using both the elasticity and phase field errors for physically meaningful, yet computationally tractable, results. They also reveal the importance of including p-refinement, which is typically not included in existing phase field literature. The above features provide a powerful and general tool for modelling fracture propagation with controlled errors and degree-of-freedom optimised meshes.more » « less
- Free Publicly Accessible Full Text
-
Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1515/cmam-2018-0230
