COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING 45 (1984) 28.5-312 NORTH-HOLLAND FINITE ELEMENT METHODS FOR LINEAR HYPERBOLIC PROBLEMS Claes JOHNSON Department zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA of Mathematics, Chalmers University of Technology, S-412 96 Giireborg, Sweden .. Uno NAVERT Flygdivisionen Saab-Scania. S-582 66 Linkiiping, Sweden Juhani PITKARANTA Department of Mathema~cs, Helsinki Uniz)ers~ty of Technology, SF-02150 Esbo 15, ~i~lan~~ Received 6 December 1982 Revised manuscript received 15 March 1983 We give a survey of some recent work by the authors on finite element methods for convection- diffusion problems and first-order linear hyperbolic problems. 0. Introduction The purpose of this paper is to give a survey of some recent work on finite element methods for convection~iffusion problems and first-order linear hyperbolic problems by the authors [9-13, 171. The present paper is an elaboration of [lo]. We shall first consider a stationary scalar linear convection-dominated convection-diffusion problem of the form -shuip*Vu+cru=f inft, (0.1) u=g on r, where 0 is a bounded domain in RN with boundary r, p = (&, . . . , &) and a are smoothly varying coefficients with IpI - 1 and E >O is a small constant. The solution u of this problem is in general not globally smooth even for smooth data zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQP f and g [20]; in general u will vary rapidly in a layer of width O(E) at the outflow boundary r+ = {x E T: n(x) * /3(x) 30) where n(x) is the outward unit normal to r at x E f. Moreover, in the limit case E = 0 where the boundary data g are only prescribed on the inflow boundary r... = {x E r: n(x) * /3(x) < 0} u will be discontinuous across the characteristic curve (streamline) x(s) given by dxfds = p(x), x(O) = x0 E r_, if e.g. g is discontinuous at x 0. If E > 0 then such a discontinuity is spread out over a layer around the characteristic x(s) of width O(~E). A ‘classical’ problem in numerical analysis is to construct, using a mesh with mesh length h not oriented to follow the characteristics, a finite difference or finite element method for (0.1) that (i) is higher-order accurate and (ii) has good stability properties without requiring h to be smaller than E. Conventional schemes for (0.1) fall into one of the following two categories 00457825/84/$3.00 @ 1984, Elsevier Science Publishers B.V. (North-Holland)