(1) A one-dimensional minimization problem and the Ritz method. (2) Weak formulation and the Galerkin method. Abstract error estimates. (3) The lemma of variations. Euler-Lagrange equations. Weak formulation, again. (4) The Ritz-Galerkin finite element method: philosophy. (5) The piecewise linear finite element space and basis functions. The linear system. (6) The piecewise linear finite elemen...

This paper is concerned with the stability of Petrov-Galerkin discretizations with application to parabolic evolution problems in space-time weak form. We will prove that the discrete inf-sup condition for an a priori fixed Petrov-Galerkin discretization is satisfied uniformly under standard approximation and smoothness conditions without any further coupling between the discrete trialand test ...

A collocation-Galerkin scheme is proposed for an initial-boundary value problem for a first order hyperbolic equation in one space dimension. The Galerkin equations satisfied by the approximating solution are obtained from a weak-weak formulation of the initial-boundary value problem. The collocation points are taken to be affine images of the roots of the Jacobian polynomials of degree r — \ o...

We consider a weak form of controllability for system that have a conserved quantity and satisfy a condition of Hörmander type. It is shown that such systems are approximately controllable under a weak growth condition for the conserved quantity. The proof of the result combines analytic tools with probabilistic arguments. A counterexample is given that shows that the growth condition is essent...

The present study considers mathematical classification of the time differential operators and then applies methods of approximation in time such as Galerkin method ( GM ), Galerkin method with weak form ( / GM WF ), Petrov-Galerkin method ( PGM ), weighted residual method (WRY ), and least squares method or process ( LSM or LSP ) to construct finite element approximations in time. A correspond...

This paper is concerned with the stability and convergence of fully discrete Galerkin methods for boundary integral equations on bounded piecewise smooth surfaces in R 3. Our theory covers equations with very general operators , provided the associated weak form is bounded and elliptic on H , for some 2 ?1; 1]. In contrast to other studies on this topic, we do not assume our meshes to be quasiu...

We consider stochastic elliptic variational inequalities of the second kind involving a bilinear form with stochastic diffusion coefficient. We prove existence and uniqueness of weak solutions, propose a stochastic Galerkin approximation of an equivalent parametric reformulation, and show equivalence to a related collocation method. Numerical experiments illustrate the efficiency of our approac...

We consider the a priori error analysis of hp-version interior penalty discontinuous Galerkin methods for second–order partial differential equations with nonnegative characteristic form under weak assumptions on the mesh design and the local finite element spaces employed. In particular, we prove a priori hp-error bounds for linear target functionals of the solution, on (possibly) anisotropic ...