arXiv subjects

Vit Dolejsi

Publications and source records attributed to Vit Dolejsi.

5 recordsLinked to original sources

arXiv2026

Adaptive domain decomposition method for time-dependent problems with applications in fluid dynamics

We deal with the numerical solution of the time-dependent partial differential equations using the adaptive space-time discontinuous Galerkin (DG) method. The discretization leads to a nonlinear algebraic system at each time level, the size of the system is varying due to mesh adaptation. A Newton-like iterative solver leads to a sequence of linear algebraic systems which are solved by GMRES solver with a domain decomposition preconditioner. Particularly, we consider additive and hybrid two-level Schwarz preconditioners which are efficient and easy to implement for DG discretization. We study the convergence of the linear solver in dependence on the number of subdomains and the number of element of the coarse grid. We propose a simplified cost model measuring the computational costs in terms of floating-point operations, the speed of computation, and the wall-clock time for communications among computer cores. Moreover, the cost model serves as a base of the presented adaptive domain decomposition method which chooses the number of subdomains and the number of element of the coarse grid in order to minimize the computational costs. The efficiency of the proposed technique is demonstrated by two benchmark problems of compressible flow simulations.

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arXiv2025

Hybrid Schwarz preconditioners for linear systems arising from hp-discontinuous Galerkin method

We deal with the numerical solution of linear elliptic problems with varying diffusion coefficient by the $hp$-discontinuous Galerkin method. We develop a two-level hybrid Schwarz preconditioner for the arising linear algebraic systems. The preconditioner is additive with respect to the local components and multiplicative with respect to the mesh levels. We derive the $hp$ spectral bound of the preconditioned operator in the form $O((H/h)(p^2/q))$, where $H$ and $h$ are the element sizes of the coarse and fine meshes, respectively, and $p$ and $q$ are the polynomial approximation degrees on the fine and coarse meshes. Further, we present a numerical study comparing the hybrid Schwarz preconditioner with the standard additive one from the point of view of the speed of convergence and also computational costs. Moreover, we investigate the convergence of both techniques with respect to the diffusivity variation and to the domain decomposition (non-)respecting the material interfaces. Finally, the combination with a $hp$-mesh adaptation for the solution of nonlinear problem demonstrates the potential of this approach.

math.NA↗

arXiv2024

Non-hydrostatic mesoscale atmospheric modeling by the anisotropic mesh adaptive discontinuous Galerkin method

We deal with non-hydrostatic mesoscale atmospheric modeling using the fully implicit space-time discontinuous Galerkin method in combination with the anisotropic $hp$-mesh adaptation technique. The time discontinuous approximation allows the treatment of different meshes at different time levels in a natural way which can significantly reduce the number of degrees of freedom. The presented approach generates a sequence of triangular meshes consisting of possible anisotropic elements and varying polynomial approximation degrees such that the interpolation error is below the given tolerance and the number of degrees of freedom at each time step is minimal. We describe the discretization of the problem together with several implementation issues related to the treatment of boundary conditions, algebraic solver and adaptive choice of the size of the time steps.The computational performance of the proposed method is demonstrated on several benchmark problems.

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arXiv2023

Goal-oriented error analysis of iterative Galerkin discretizations for nonlinear problems including linearization and algebraic errors

We consider the goal-oriented error estimates for a linearized iterative solver for nonlinear partial differential equations. For the adjoint problem and iterative solver we consider, instead of the differentiation of the primal problem, a suitable linearization which guarantees the adjoint consistency of the numerical scheme. We derive error estimates and develop an efficient adaptive algorithm which balances the errors arising from the discretization and use of iterative solvers. Several numerical examples demonstrate the efficiency of this algorithm.

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arXiv2020

Goal-oriented anisotropic $hp$-adaptive discontinuous Galerkin method for the Euler equations

We deal with the numerical solution of the compressible Euler equations with the aid of the discontinuous Galerkin (DG) method with focus on the goal-oriented error estimates and adaptivity. We analyze the adjoint consistency of the DG scheme where the dual problem is not formulated by the differentiation of the DG form and the target functional but using a suitable linearization of the nonlinear forms. Further, we present the goal-oriented anisotropic $hp$-mesh adaptation technique for the Euler equations. The theoretical results are supported by numerical experiments.

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