In this paper we consider a direct hypersingular integral approach to solve harmonic problems with nonlinear boundary conditions by using a practical variant of the Galerkin boundary element method. The proposed approach provides an almost optimal balance between the order of convergence and the numerical effort of work to compute the approximate solution. Numerical examples confirm the theoret...

Modifications to the standard finite element mass matrix are considered with the aim of preserving the positivity of the discrete solution. The approach is used in connection with calculating the initial time derivative values for parabolic equations and in connection with nonlinear Petrov-Galerkin schemes for hyperbolic equations in one space dimension. The extension of the ideas to unstructur...

We generalize the a posteriori techniques for the linear heat equation in [Ver03] to the case of the nonlinear parabolic p-Laplace problem thereby proving reliable and efficient a posteriori error estimates for a fully discrete implicite Euler Galerkin finite element scheme. The error is analyzed using the so-called quasi-norm and a related dual error expression. This leads to equivalence of th...

We investigate a macro-element variant of the hybridized discontinuous Galerkin (HDG) method, using patches standard simplicial elements that can have non-matching interfaces. Coupled via HDG technique, our method enables local refinement by uniform subdivision each macro-element. By enforcing one spatial discretization for all macro-elements, we arrive at problems per are embarrassingly parall...

The problem of constraint preservation for discretizations of nonlinear PDEs is addressed in the example of the hyperbolic Yang–Mills equations in temporal gauge. These equations preserve a nonlinear divergence field analogous to the electric charge for Maxwell’s equations. We introduce and discuss several discretizations of these equations on finite element spaces of Lie algebra valued differe...

In this paper, we consider the constructive a priori error estimates for a full discrete numerical solution of the heat equation. Our method is based on the finite element Galerkin method with an interpolation in time that uses the fundamental solution for semidiscretization in space. The present estimates play an essential role in the numerical verification method of exact solutions for the no...