We consider the initial boundary value problem for the homogeneous heat equation, with homogeneous Dirichlet boundary conditions. By the maximum principle the solution is nonnegative for positive time if the initial data are nonnegative. We study to what extent this property carries over to some piecewise linear finite element discretizations, namely the Standard Galerkin method, the Lumped Mas...

In this paper we prove convergence and error estimates for the socalled 3-field formulation using piecewise linear finite elements stabilized with boundary bubbles. Optimal error bounds are proved in L2 and in the broken H1 norm for the internal variable u, and in suitable weighted L2 norms for the other variables λ and ψ.

We analyze an immersed interface finite element method based on linear polynomials on noninterface triangular elements and piecewise linear polynomials on interface triangular elements. The flux jump condition is weakly enforced on the smooth interface. Optimal error estimates are derived in the broken H1-norm and L2-norm.

We prove that a discrete maximum principle holds for continuous piecewise linear finite element approximations for the Poisson equation with the Dirichlet boundary condition also under a condition of the existence of some obtuse internal angles between faces of terahedra of triangulations of a given space domain. This result represents a weakened form of the acute type condition for the three-d...

We consider a continuous-time optimization method based on dynamical system, where massive particle starting at rest moves in the conservative force field generated by objective function, without any kind of friction. formulate restart criterion mean dissipation kinetic energy, and we prove global convergence result for strongly-convex functions. Using Symplectic Euler discretization scheme, ob...