82 research outputs found
A Two-Level Method for Mimetic Finite Difference Discretizations of Elliptic Problems
We propose and analyze a two-level method for mimetic finite difference
approximations of second order elliptic boundary value problems. We prove that
the two-level algorithm is uniformly convergent, i.e., the number of iterations
needed to achieve convergence is uniformly bounded independently of the
characteristic size of the underling partition. We also show that the resulting
scheme provides a uniform preconditioner with respect to the number of degrees
of freedom. Numerical results that validate the theory are also presented
On the Virtual Element Method for Topology Optimization on polygonal meshes: a numerical study
It is well known that the solution of topology optimization problems may be
affected both by the geometric properties of the computational mesh, which can
steer the minimization process towards local (and non-physical) minima, and by
the accuracy of the method employed to discretize the underlying differential
problem, which may not be able to correctly capture the physics of the problem.
In light of the above remarks, in this paper we consider polygonal meshes and
employ the virtual element method (VEM) to solve two classes of paradigmatic
topology optimization problems, one governed by nearly-incompressible and
compressible linear elasticity and the other by Stokes equations. Several
numerical results show the virtues of our polygonal VEM based approach with
respect to more standard methods
Discontinuous Galerkin approximation of linear parabolic problems with dynamic boundary conditions
In this paper we propose and analyze a Discontinuous Galerkin method for a
linear parabolic problem with dynamic boundary conditions. We present the
formulation and prove stability and optimal a priori error estimates for the
fully discrete scheme. More precisely, using polynomials of degree on
meshes with granularity along with a backward Euler time-stepping scheme
with time-step , we prove that the fully-discrete solution is bounded
by the data and it converges, in a suitable (mesh-dependent) energy norm, to
the exact solution with optimal order . The sharpness of the
theoretical estimates are verified through several numerical experiments
High order discontinuous Galerkin methods on surfaces
We derive and analyze high order discontinuous Galerkin methods for
second-order elliptic problems on implicitely defined surfaces in
. This is done by carefully adapting the unified discontinuous
Galerkin framework of Arnold et al. [2002] on a triangulated surface
approximating the smooth surface. We prove optimal error estimates in both a
(mesh dependent) energy norm and the norm.Comment: 23 pages, 2 figure
Iterative solution to the biharmonic equation in mixed form discretized by the Hybrid High-Order method
We consider the solution to the biharmonic equation in mixed form discretized
by the Hybrid High-Order (HHO) methods. The two resulting second-order elliptic
problems can be decoupled via the introduction of a new unknown, corresponding
to the boundary value of the solution of the first Laplacian problem. This
technique yields a global linear problem that can be solved iteratively via a
Krylov-type method. More precisely, at each iteration of the scheme, two
second-order elliptic problems have to be solved, and a normal derivative on
the boundary has to be computed. In this work, we specialize this scheme for
the HHO discretization. To this aim, an explicit technique to compute the
discrete normal derivative of an HHO solution of a Laplacian problem is
proposed. Moreover, we show that the resulting discrete scheme is well-posed.
Finally, a new preconditioner is designed to speed up the convergence of the
Krylov method. Numerical experiments assessing the performance of the proposed
iterative algorithm on both two- and three-dimensional test cases are
presented
A DG-VEM method for the dissipative wave equation
A novel space-time discretization for the (linear) scalar-valued dissipative
wave equation is presented. It is a structured approach, namely, the
discretization space is obtained tensorizing the Virtual Element (VE)
discretization in space with the Discontinuous Galerkin (DG) method in time. As
such, it combines the advantages of both the VE and the DG methods. The
proposed scheme is implicit and it is proved to be unconditionally stable and
accurate in space and time
A cVEM-DG space-time method for the dissipative wave equation
A novel space-time discretization for the (linear) scalar-valued dissipative wave equation is presented. It is a structured approach, namely, the discretization space is obtained tensorizing the Virtual Element (VE) discretization in space with the Discontinuous Galerkin (DG) method in time. As such, it combines the advantages of both the VE and the DG methods. The proposed scheme is implicit and it is proved to be unconditionally stable and accurate in space and time
- …