After discretisation of the partial differential equations from mechanics one usually obtains large systems of (non)linear equations. Their efficient solution requires the use of fast iterative methods. Multi-grid iterations are able to solve linear and nonlinear systems with a rather fast rate of convergence, provided the problem is of elliptic type. The contribution describes the basic constr...

The convergence of finite element methods for elliptic and parabolic partial differential equations is well-established if source terms are sufficiently smooth. Noting that finite element computation is easily implemented even when the source terms are measure-valued — for instance, modeling point sources by Dirac delta distributions — we prove new convergence order results in two and three dim...

The stochastic Galerkin finite element method (SGFEM) provides an efficient alternative to traditional sampling methods for the numerical solution of linear elliptic partial differential equations ...

The paper considers a class of parametric elliptic partial differential equations (PDEs), where the coefficients and right-hand side function depend on infinitely many (uncertain) parameters. W...

This paper develops and analyzes an efficient numerical method for solving elliptic partial differential equations, where the diffusion coefficients are random perturbations of deterministic diffusion coefficients. The method is based upon a multimodes representation of the solution as a power series of the perturbation parameter, and the Monte Carlo technique for sampling the probability space...

we prove the existence of steady 2-dimensional flows, containing a bounded vortex, and approaching a uniform flow at infinity. the data prescribed is the rearrangement class of the vorticity field. the corresponding stream function satisfies a semilinear elliptic partial differential equation. the result is proved by maximizing the kinetic energy over all flows whose vorticity fields are rearra...

A double sub-equation method is presented for constructing complexiton solutions of nonlinear partial differential equations (PDEs). The main idea of the method is to take full advantage of two solvable ordinary differential equations with different independent variables. With the aid of Maple, one can obtain both complexiton solutions, combining elementary functions and the Jacobi elliptic fun...

We present an implementation of Böhmer’s finite element method for fully nonlinear elliptic partial differential equations on convex polygonal domains, based on a modified Argyris element and BernsteinBézier techniques. Our numerical experiments for several test problems, involving the classical Monge-Ampère equation and an unconditionally elliptic equation, confirm the convergence and error bo...

In this paper we study the Neumann problem for a type of fully nonlinear second order elliptic partial differential equations on domains in Cn without any curvature assumptions domain.

We investigate the conditions under which residual-based variational multiscale methods are adjoint, or dual, consistent for model hyperbolic and elliptic partial differential equations. In particular, while many residual-based variational multiscale stabilizations are adjoint consistent for hyperbolic problems and finite-element spaces, only a few are adjoint consistent for elliptic problems.