Galerkin approximations to solutions of a Cauchy-Dirichlet problem governed by the generalized porous medium equation

On the existence for solutions of evolutionary equations, there are many works and methods 1–7 . For example, the fixed principle 1, 3–5, 7 and Galerkin approximations 2, 6 . They are very classical methods to prove existence and uniqueness. Generally speaking, there are four solution concepts. That is, weak solution, mild solution, strong solution, and classical solution. We can obtain differe...

In this paper, we consider discontinuous Galerkin approximations to the solution of Naghdi arches and show how to post-process them in an element-by-element fashion to obtain a far better approximation. Indeed, we prove that, if polynomials of degree k are used, the postprocessed approximation converges with order 2k+1 in the L2-norm throughout the domain. This has to be contrasted with the fac...

We consider the Galerkin finite element approximation of an elliptic Dirichlet boundary control model problem governed by the Laplacian operator. The functional theoretical setting of this problem uses L2 controls and a “very weak” formulation of the state equation. However, the corresponding finite element approximation uses standard continuous trial and test functions. For this approximation,...

Let u^ be a Ritz-Galerkin approximation, corresponding to the solution u of an elliptic boundary value problem, which is based on a uniform subdivision in the interior of the domain. In this paper we show that by "averaging" the values of Uu in the neighborhood of a point x we may (for a wide class of problems) construct an approximation to u(x) which is often a better approximation than UyAx) ...

Galerkin approximations to solutions of a Cauchy-Dirichlet problem governed by the generalized porous medium equation

We consider an unbiased approximation of stochastic Navier-Stokes equation driven by spatial white noise. This perturbation is unbiased in that the expectation of a solution of the perturbed equation solves the deterministic Navier-Stokes equation. The nonlinear term can be characterized as the highest stochastic order approximation of the original nonlinear term u∇u. We investigate the analyti...

As a model benchmark problem for this study we consider a highly singular transmission type eigenvalue problem which we study in detail both analytically as well as numerically. In order to justify our claim of cluster robust and highly accurate approximation of a selected groups of eigenvalues and associated eigenfunctions, we give a new analysis of a class of direct residual eigenspace/vector...