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Register a new userMultigrid algorithms for $hp$-version Interior Penalty Discontinuous Galerkin methods on polygonal and polyhedral meshes
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In this paper we analyze the convergence properties of two-level and W-cycle multigrid solvers for the numerical solution of the linear system of equations arising from hp-version symmetric interior penalty discontinuous Galerkin discretizations of second-order elliptic partial differential equations on polygonal/polyhedral meshes. We prove that the two-level method converges uniformly with respect to the granularity of the grid and the polynomial approximation degree p, provided that the number of smoothing steps, which depends on p, is chosen sufficiently large. An analogous result is obtained for the W-cycle multigrid algorithm, which is proved to be uniformly convergent with respect to the mesh size, the polynomial approximation degree, and the number of levels, provided the number of smoothing steps is chosen sufficiently large. Numerical experiments are presented which underpin the theoretical predictions; moreover, the proposed theoretical assumptions are not fully satisfied.
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In this paper we analyse the convergence properties of V-cycle multigrid algorithms for the numerical solution of the linear system of equations arising from discontinuous Galerkin discretization of second-order elliptic partial differential equations on polytopal meshes. Here, the sequence of spaces that stands at the basis of the multigrid scheme is possibly non nested and is obtained based on employing agglomeration with possible edge/face coarsening. We prove that the method converges uniformly with respect to the granularity of the grid and the polynomial approximation degree p, provided that the number of smoothing steps, which depends on p, is chosen sufficiently large.
In this paper, the optimal choice of the interior penalty parameter of the discontinuous Galerkin finite element methods for both the elliptic problems and the Biots systems are studied by utilizing the neural network and machine learning. It is crucial to choose the optimal interior penalty parameter, which is not too small or not too large for the stability, robustness, and efficiency of the numerical discretized solutions. Both linear regression and nonlinear artificial neural network methods are employed and compared using several numerical experiments to illustrate the capability of our proposed computational framework. This framework is an integral part of a developing automated numerical simulation platform because it can automatically identify the optimal interior penalty parameter. Real-time feedback could also be implemented to update and improve model accuracy on the fly.
This paper proposes and analyzes two fully discrete mixed interior penalty discontinuous Galerkin (DG) methods for the fourth order nonlinear Cahn-Hilliard equation. Both methods use the backward Euler method for time discretization and interior penalty discontinuous Galerkin methods for spatial discretization. They differ from each other on how the nonlinear term is treated, one of them is based on fully implicit time-stepping and the other uses the energy-splitting time-stepping. The primary goal of the paper is to prove the convergence of the numerical interfaces of the DG methods to the interface of the Hele-Shaw flow. This is achieved by establishing error estimates that depend on $epsilon^{-1}$ only in some low polynomial orders, instead of exponential orders. Similar to [14], the crux is to prove a discrete spectrum estimate in the discontinuous Galerkin finite element space. However, the validity of such a result is not obvious because the DG space is not a subspace of the (energy) space $H^1$ and it is larger than the finite element space. This difficult is overcome by a delicate perturbation argument which relies on the discrete spectrum estimate in the finite element space proved in cite{Feng_Prohl04}. Numerical experiment results are also presented to gauge the theoretical results and the performance of the proposed fully discrete mixed DG methods.
Finite element simulations have been used to solve various partial differential equations (PDEs) that model physical, chemical, and biological phenomena. The resulting discretized solutions to PDEs often do not satisfy requisite physical properties, such as positivity or monotonicity. Such invalid solutions pose both modeling challenges, since the physical interpretation of simulation results is not possible, and computational challenges, since such properties may be required to advance the scheme. We, therefore, consider the problem of computing solutions that preserve these structural solution properties, which we enforce as additional constraints on the solution. We consider in particular the class of convex constraints, which includes positivity and monotonicity. By embedding such constraints as a postprocessing convex optimization procedure, we can compute solutions that satisfy general types of convex constraints. For certain types of constraints (including positivity and monotonicity), the optimization is a filter, i.e., a norm-decreasing operation. We provide a variety of tests on one-dimensional time-dependent PDEs that demonstrate the methods efficacy, and we empirically show that rates of convergence are unaffected by the inclusion of the constraints.
We design and analyze a coupling of a discontinuous Galerkin finite element method with a boundary element method to solve the Helmholtz equation with variable coefficients in three dimensions. The coupling is realized with a mortar variable that is related to an impedance trace on a smooth interface. The method obtained has a block structure with nonsingular subblocks. We prove quasi-optimality of the $h$- and $
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