Chapter 3 Conforming Finite Element Methods 3 . 1 Foundations 3 . 1 . 1 Ritz - Galerkin Method
نویسنده
چکیده
Let V be a Hilbert space, a(·, ·) : V × V → lR a bounded, V-elliptic bilinear form and : V → lR a bounded linear functional. We want to approximate the variational equation: Find u ∈ V such that (3.1) a(u, v) = (v) , v ∈ V. We recall that the Lax-Milgram Lemma ensures the existence and uniqueness of a solution of (3.1). Now, given a finite-dimensional subspace V h ⊂ V , dim V h = n h , the Ritz-Galerkin method is to approximate (3.1) by its restriction to V h : Find u h ∈ V h such that (3.2) a(u h , v h) = (v h) , v h ∈ V h .
منابع مشابه
AMSC 614 : Mathematics of the Finite Element Method
(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...
متن کاملA Synthesis of A Posteriori Error Estimation Techniques for Conforming, Non-Conforming and Discontinuous Galerkin Finite Element Methods
A posteriori error estimation for conforming, non-conforming and discontinuous finite element schemes are discussed within a single framework. By dealing with three ostensibly different schemes under the same umbrella, the same common underlying principles at work in each case are highlighted leading to a clearer understanding of the issues involved. The ideas are presented in the context of pi...
متن کاملBlock Jacobi for discontinuous Galerkin discretizations: no ordinary Schwarz methods
For classical discretizations of elliptic partial differential equations, like conforming finite elements or finite differences, block Jacobi methods are equivalent to classical Schwarz methods with minimal overlap, see for example [4]. This is different when the linear system (1) is obtained using DG methods. Our paper is organized as follows: in section 2 we describe several DG methods for li...
متن کاملA Sparse Grid Discretization with Variable Coefficient in High Dimensions
We present a Ritz-Galerkin discretization on sparse grids using pre-wavelets, which allows to solve elliptic differential equations with variable coefficients for dimension d = 2, 3 and higher dimensions d > 3. The method applies multilinear finite elements. We introduce an efficient algorithm for matrix vector multiplication using a Ritz-Galerkin discretization and semi-orthogonality. This alg...
متن کاملA non-conforming discontinuous Galerkin method for solving Maxwell's equations
This paper reviews the main features of a high-order nondissipative discontinuous Galerkin (DG) method recently investigated in [1]-[3] for solving Maxwell’s equations on non-conforming simplex meshes. The proposed method combines a centered approximation for the numerical fluxes at inter element boundaries, with either a secondorder or a fourth-order leap-frog time integration scheme. Moreover...
متن کامل