We developed a reduced order model (ROM) using the proper orthogonal decomposition (POD) to compute efficiently the labyrinth and spot like patterns of the FitzHugh-Nagumo (FNH) equation. The FHN equation is discretiz...
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ISBN:
(纸本)9783319399294;9783319399270
We developed a reduced order model (ROM) using the proper orthogonal decomposition (POD) to compute efficiently the labyrinth and spot like patterns of the FitzHugh-Nagumo (FNH) equation. The FHN equation is discretized in space by the discontinuous Galerkin (dG) method and in time by the backward Euler method. Applying POD-DEIM (discrete empirical interpolation method) to the full order model (FOM) for different values of the parameter in the bistable nonlinearity, we show that using few POD and DEIM modes, the patterns can be computed accurately. Due to the local nature of the dG discretization, the POD-DEIM requires less number of connected nodes than continuous finite element for the nonlinear terms, which leads to a significant reduction of the computational cost for dG POD-DEIM.
In this paper we consider homogeneous Dirichlet problem for the Lame system with singularity caused by the reentrant corner to the boundary of the twodimensional domain. For this problem we define the solution as a R-...
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ISBN:
(纸本)9783319399294;9783319399270
In this paper we consider homogeneous Dirichlet problem for the Lame system with singularity caused by the reentrant corner to the boundary of the twodimensional domain. For this problem we define the solution as a R-v -generalized one;we state its existence and uniqueness in the weighted set W-2,v(1) (Omega,delta). On the basis of the R-v -generalized solution we construct weighted finite element method. We prove that the approximate solution converges to the exact one with the rate O(h) in the norm of W-2,v(1) (Omega) and results of numerical experiments are presented.
The numerical approximation of partial differential equations (PDEs) posed on complicated geometries, which include a large number of small geometrical features or microstructures, represents a challenging computation...
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ISBN:
(纸本)9783319416403;9783319416380
The numerical approximation of partial differential equations (PDEs) posed on complicated geometries, which include a large number of small geometrical features or microstructures, represents a challenging computational problem. Indeed, the use of standard mesh generators, employing simplices or tensor product elements, for example, naturally leads to very fine finite element meshes, and hence the computational effort required to numerically approximate the underlying PDE problem may be prohibitively expensive. As an alternative approach, in this article we present a review of composite/agglomerated discontinuous Galerkin finite element methods (DGFEMs) which employ general polytopic elements. Here, the elements are typically constructed as the union of standard element shapes;in this way, the minimal dimension of the underlying composite finite element space is independent of the number of geometrical features. In particular, we provide an overview of hp-version inverse estimates and approximation results for general polytopic elements, which are sharp with respect to element facet degeneration. On the basis of these results, a priori error bounds for the hp-DGFEM approximation of both second-order elliptic and first-order hyperbolic PDEs will be derived. Finally, we present numerical experiments which highlight the practical application of DGFEMs on meshes consisting of general polytopic elements.
In this paper we extend the study of (dimension) adaptive sparse grids by building a lattice framework around projections onto hierarchical surpluses. Using this we derive formulas for the explicit calculation of comb...
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ISBN:
(数字)9783319282626
ISBN:
(纸本)9783319282626;9783319282602
In this paper we extend the study of (dimension) adaptive sparse grids by building a lattice framework around projections onto hierarchical surpluses. Using this we derive formulas for the explicit calculation of combination coefficients, in particular providing a simple formula for the coefficient update used in the adaptive sparse grids algorithm. Further, we are able to extend error estimates for classical sparse grids to adaptive sparse grids. Multi-variate extrapolation has been well studied in the context of sparse grids. This too can be studied within the adaptive sparse grids framework and doing so leads to an adaptive extrapolation algorithm.
We develop an adaptive method of time layers with a linearly implicit Rosenbrock method as time integrator and symmetric interior penalty Galerkin method for space discretization for the advective Allen-Cahn equation ...
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ISBN:
(纸本)9783319399294;9783319399270
We develop an adaptive method of time layers with a linearly implicit Rosenbrock method as time integrator and symmetric interior penalty Galerkin method for space discretization for the advective Allen-Cahn equation with a non-divergence-free velocity field. Numerical simulations for advection-dominated problems demonstrate the accuracy and efficiency of the adaptive algorithm for resolving the sharp layers occurring in interface problems with small surface tension.
The problem in dimensionless variables reduces to three systems of equations for the stream function and the electric potential in three regions of a strip (the rectangular domain bounded by the electrodes and two hal...
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ISBN:
(纸本)9783319399294;9783319399270
The problem in dimensionless variables reduces to three systems of equations for the stream function and the electric potential in three regions of a strip (the rectangular domain bounded by the electrodes and two half-strips). The singular integral equations obtained from the integral representation of the solutions and the matching conditions are disctretized and a linear system of algebraic equations is obtained. The velocity, the electric field and the generator power are calculated.
This paper gives numerical experiments for the Finite Element Heterogeneous Multiscale Method applied to problems in linear elasticity, which has been analyzed in Abdulle (MathModels Methods Appl Sci 16: 615-635, 2006...
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ISBN:
(纸本)9783319399294;9783319399270
This paper gives numerical experiments for the Finite Element Heterogeneous Multiscale Method applied to problems in linear elasticity, which has been analyzed in Abdulle (MathModels Methods Appl Sci 16: 615-635, 2006). The main results for the FE-HMM a priori errors are stated and their sharpness are verified though numerical experiments.
We consider second order hyperbolic equation with nonlocal integral boundary conditions. We study the spectrum of the weighted difference operator for the formulated problem. Using the characteristic function we inves...
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ISBN:
(纸本)9783319399294;9783319399270
We consider second order hyperbolic equation with nonlocal integral boundary conditions. We study the spectrum of the weighted difference operator for the formulated problem. Using the characteristic function we investigate the spectrum of the transition matrix of the three-layered finite difference scheme and obtain spectral stability conditions subject to boundary parameters gamma(0), gamma(1) and piecewise constant weight functions.
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