86 research outputs found
The nonconforming virtual element method for eigenvalue problems
We analyse the nonconforming Virtual Element Method (VEM) for the
approximation of elliptic eigenvalue problems. The nonconforming VEM allow to
treat in the same formulation the two- and three-dimensional case.We present
two possible formulations of the discrete problem, derived respectively by the
nonstabilized and stabilized approximation of the L^2-inner product, and we
study the convergence properties of the corresponding discrete eigenvalue
problem. The proposed schemes provide a correct approximation of the spectrum,
in particular we prove optimal-order error estimates for the eigenfunctions and
the usual double order of convergence of the eigenvalues. Finally we show a
large set of numerical tests supporting the theoretical results, including a
comparison with the conforming Virtual Element choice
Conforming and nonconforming virtual element methods for elliptic problems
We present in a unified framework new conforming and nonconforming Virtual
Element Methods (VEM) for general second order elliptic problems in two and
three dimensions. The differential operator is split into its symmetric and
non-symmetric parts and conditions for stability and accuracy on their discrete
counterparts are established. These conditions are shown to lead to optimal
- and -error estimates, confirmed by numerical experiments on a set
of polygonal meshes. The accuracy of the numerical approximation provided by
the two methods is shown to be comparable
The Discrete Duality Finite Volume Method for Convection Diffusion Problems
In this paper we extend the Discrete Duality Finite Volume (DDFV) formulation to the steady convection-diffusion equation. The discrete gradients defined in DDFV are used to define a cell-based gradient for the control volumes of both the primal and dual meshes, in order to achieve a higher-order accurate numerical flux for the convection term. A priori analysis is carried out to how convergence of the approximation and a global first-order convergence rate is derived. The theoretical results are confirmed bysome numerical experiments
A CeVeFE DDFV scheme for discontinuous anisotropic permeability tensors
International audienceIn this work we derive a formulation for discontinuous diffusion tensor for the Discrete Duality Finite Volume (DDFV) framework that is exact for affine solutions. In fact, DDFV methods can naturally handle anisotropic or non-linear problems on general distorded meshes. Nonetheless, a special treatment is required when the diffusion tensor is discontinuous across an internal interfaces shared by two control volumes of the mesh. In such a case, two different gradients are considered in the two subdiamonds centered at that interface and the flux conservation is imposed through an auxiliary variable at the interface
A decision-making machine learning approach in Hermite spectral approximations of partial differential equations
The accuracy and effectiveness of Hermite spectral methods for the numerical
discretization of partial differential equations on unbounded domains, are
strongly affected by the amplitude of the Gaussian weight function employed to
describe the approximation space. This is particularly true if the problem is
under-resolved, i.e., there are no enough degrees of freedom. The issue becomes
even more crucial when the equation under study is time-dependent, forcing in
this way the choice of Hermite functions where the corresponding weight depends
on time. In order to adapt dynamically the approximation space, it is here
proposed an automatic decision-making process that relies on machine learning
techniques, such as deep neural networks and support vector machines. The
algorithm is numerically tested with success on a simple 1D problem, but the
main goal is its exportability in the context of more serious applications.Comment: 22 pages, 4 figure
The Tensor-Train Stochastic Finite Volume Method for Uncertainty Quantification
The stochastic finite volume method offers an efficient one-pass approach for
assessing uncertainty in hyperbolic conservation laws. Still, it struggles with
the curse of dimensionality when dealing with multiple stochastic variables. We
introduce the stochastic finite volume method within the tensor-train framework
to counteract this limitation. This integration, however, comes with its own
set of difficulties, mainly due to the propensity for shock formation in
hyperbolic systems. To overcome these issues, we have developed a
tensor-train-adapted stochastic finite volume method that employs a global WENO
reconstruction, making it suitable for such complex systems. This approach
represents the first step in designing tensor-train techniques for hyperbolic
systems and conservation laws involving shocks
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