Theory of differential equations in discontinuous piecewise‐defined functions

Abstract

AbstractA truly general and systematic theory of finite element methods (FEM) should be formulated using, as trial and test functions, piecewise‐defined functions that can be fully discontinuous across the internal boundary, which separates the elements from each other. Some of the most relevant work addressing such formulations is contained in the literature on discontinuous Galerkin (dG) methods and on Trefftz methods. However, the formulations of partial differential equations in discontinuous functions used in both of those fields are indirect approaches, which are based on the use of Lagrange multipliers and mixed methods, in the case of dG methods, and the frame, in the case of Trefftz method. This article addresses this problem from a different point of view and proposes a theory, formulated in discontinuous piecewise‐defined functions, which is direct and systematic, and furthermore it avoids the use of Lagrange multipliers or a frame, while mixed methods are incorporated as particular cases of more general results implied by the theory. When boundary value problems are formulated in discontinuous functions, well‐posed problems are boundary value problems with prescribed jumps (BVPJ), in which the boundary conditions are complemented by suitable jump conditions to be satisfied across the internal boundary of the domain‐partition. One result that is presented in this article shows that for elliptic equations of order 2m, with m ≥ 1, the problem of establishing conditions for existence of solution for the BVPJ reduces to that of the “standard boundary value problem,” without jumps, which has been extensively studied. Actually, this result is an illustration of a more general one that shows that the same happens for any differential equation, or system of such equations that is linear, independently of its type and with possibly discontinuous coefficients. This generality is achieved by means of an algebraic framework previously developed by the author and his collaborators. A fundamental ingredient of this algebraic formulation is a kind of Green's formulas that simplify many problems (some times referred to as Green‐Herrera formulas). An important practical implication of our approach is worth mentioning: “avoiding the introduction of the Lagrange multipliers, or the ‘frame’ in the case of Trefftz‐methods, significantly reduces the number of degrees of freedom to be dealt with.” © 2006 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2007