Abstract We present a new domain decomposition algorithm for the parallel finite element solution of elliptic partial differential equations. As with most parallel domain decomposition methods each processor is assigned one or more subdomains and an iteration is devised which allows the processors to solve their own subproblem(s) concurrently. The novel feature of this algorithm however is that each of these subproblems is defined over the entire domain—although the vast majority of the degrees of freedom for each subproblem are associated with a single subdomain (owned by the corresponding processor). This ensures that a global mechanism is contained within each of the subproblems tackled and so no separate coarse grid solve is required in order to achieve rapid convergence of the overall iteration. Furthermore, by following the paradigm introduced in 15 , it is demonstrated that this domain decomposition solver may be coupled easily with a conventional mesh refinement code, thus allowing the accuracy, reliability and efficiency of mesh adaptivity to be utilized in a well load‐balanced manner. Finally, numerical evidence is presented which suggests that this technique has significant potential, both in terms of the rapid convergence properties and the efficiency of the parallel implementation. Copyright © 2001 John Wiley & Sons, Ltd.

A hybrid finite-element method to discretize the continuity equation in semiconductor device simulation is given. Within each element of a finite element discretization, the current is uniquely determined by nodal values of the density and the potential. The authors use the integrability condition for a system of partial differential equations to obtain the equations that determine the current within the element. They then satisfy the continuity in the current flow across interelement boundaries in a weak sense. They have found that the method works in any dimension and for (d-dimensional) simplexes as well as for quadrilaterals, bricks, prisms, and so on, although they have no proof that it will not break down in particular cases.>

In this paper, we present an overview of the physical principles and numerical methods used to solve the coupled system of nonlinear partial differential equations that model the transient behavior of silicon VLSI device structures. We also describe how the same techniques are applicable to circuit simulation. A composite linear multistep formula is introduced as the time-integration scheme. Newton-iterative methods are exploited to solve the nonlinear equations that arise at each time step. We also present a simple data structure for nonsymmetric matrices with symmetric nonzero structures that facilitates iterative or direct methods with substantial efficiency gains over other storage schemes. Several computational examples, including a CMOS latchup problem, are presented and discussed.

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