We discuss the numerical solution of differential equations of fractional order with discontinuous right–hand side. Problems of this kind arise, for instance, in sliding mode control. After applying a set–valued regularization, the behavior of some generalizations of the implicit Euler method is investigated. We show that the scheme in the family of fractional Adams methods possesses the same c...

A new iterative algorithm for solving initial data inverse problems from partial observations has been recently proposed in Ramdani, Tucsnak and Weiss [24]. Based on the concept of observers (also called Luenberger observers), this algorithm covers a large class of abstract evolution PDE’s. In this paper, we are concerned with the convergence analysis of this algorithm. More precisely, we provi...

This paper formulates and analyzes a weak solution to the coupling of time-dependent Navier-Stokes flow with Darcy flow under certain boundary conditions, one of them being the Beaver-Joseph-Saffman law on the interface. Existence and a priori estimates for the weak solution are shown under additional regularity assumptions. We introduce a fully discrete scheme with the unknowns being the Navie...

This is the first in a series of papers on implementing a discontinuous Galerkin method as a MATLAB /GNUOctave toolbox. The main goal is the development of techniques that deliver optimized computational performance combined with a compact, user-friendly interface. Our implementation relies on fully vectorized matrix / vector operations and is carefully documented; in addition, a direct mapping...

A multi-component reactive transport model, coupled to an existing thermal, hydraulic and mechanical model is investigated. The model is based on conservation of mass/energy principles for the flow and stress strain equilibrium for the mechanical behaviour. The resultant model is coupled to a geochemical model in order to capture geochemical interactions. Numerically, the Galerkin finite elemen...

We consider initial/boundary value problems for the subdiffusion and diffusionwave equations involving a Caputo fractional derivative in time. We develop two fully discrete schemes based on the piecewise linear Galerkin finite element method in space and convolution quadrature in time with the generating function given by the backward Euler method/second-order backward difference method, and es...

We consider a time-dependent and a steady linear convection-diffusion equation. These equations are approximately solved by a combined finite element – finite volume method: the diffusion term is discretized by Crouzeix-Raviart piecewise linear finite elements on a triangular grid, and the convection term by upwind barycentric finite volumes. In the unsteady case, the implicit Euler method is u...

This paper deals with the numerical approximation of the solution of 1D parabolic singularly perturbed problems of reaction–diffusion type. The numerical method combines the standard implicit Euler method on a uniform mesh to discretize in time and a HODIE compact fourth order finite difference scheme to discretize in space, which is defined on a priori special meshes condensing the grid points...

Implicit integration methods have contributed to large performance enhancements in the field of simulation of particle-system mechanical models. While Backward Euler and BDF-2 methods are now widely used for cloth simulation applications, the Implicit Midpoint method is often overlooked, because of its poor stability properties. It is however as simple to implement as Backward Euler, and offers...