TY - JOUR

T1 - A proof of convergence for the combination technique for the Laplace equation using tools of symbolic computation

AU - Bungartz, H.

AU - Griebel, M.

AU - Röschke, D.

AU - Zenger, C.

N1 - Funding Information:
??Supported by the Bayerische Forschungsstiftung via FORTWIHR - Bayerischer Forschungsverbund fur technisch-wissenschaftliches Hochleistungsrechnen, and by the Deutsche Forschungsgemeinschaft in its Sonderforschungsbereich 342. * Corresponding author, Tel. : 49 89 289 22018; fax: 49 89 289 22022; E-mail: [email protected].

PY - 1996/11

Y1 - 1996/11

N2 - For a simple model problem - the Laplace equation on the unit square with a Dirichlet boundary function vanishing for x = 0, x = 1 and y = 1, and equaling some suitable g (x) for y = 0 - we present a proof of convergence for the so-called combination technique, a modern, efficient and easily parallelizable sparse grid solver for elliptic partial differential equations that recently gained importance in fields of applications like computational fluid dynamics. For full square grids with meshwidth h and O(h-2) grid points, the order O(h2) of the discretization error was shown in (Hofman, 1967), if g(x) ∈ C2[0, 1]. In this paper, we show that the error of the solution produced by the combination technique on a sparse grid with only O((h-1 log2(h-1)) grid points is of the order O(h2 log2(h-1)), if g ∈ C4[0, 1], and g(0) = g(1) = g″(0) = g″(1) = 0. The crucial task of the proof, i.e. the determination of the discretization error on rectangular grids with arbitrary meshwidths in each coordinate direction, is supported by an extensive and interactive use of Maple.

AB - For a simple model problem - the Laplace equation on the unit square with a Dirichlet boundary function vanishing for x = 0, x = 1 and y = 1, and equaling some suitable g (x) for y = 0 - we present a proof of convergence for the so-called combination technique, a modern, efficient and easily parallelizable sparse grid solver for elliptic partial differential equations that recently gained importance in fields of applications like computational fluid dynamics. For full square grids with meshwidth h and O(h-2) grid points, the order O(h2) of the discretization error was shown in (Hofman, 1967), if g(x) ∈ C2[0, 1]. In this paper, we show that the error of the solution produced by the combination technique on a sparse grid with only O((h-1 log2(h-1)) grid points is of the order O(h2 log2(h-1)), if g ∈ C4[0, 1], and g(0) = g(1) = g″(0) = g″(1) = 0. The crucial task of the proof, i.e. the determination of the discretization error on rectangular grids with arbitrary meshwidths in each coordinate direction, is supported by an extensive and interactive use of Maple.

KW - Combination technique

KW - Elliptic partial differential equations

KW - Order of discretization error

KW - Sparse grids

KW - Symbolic computation

UR - http://www.scopus.com/inward/record.url?scp=0030290782&partnerID=8YFLogxK

U2 - 10.1016/S0378-4754(96)00036-5

DO - 10.1016/S0378-4754(96)00036-5

M3 - Article

AN - SCOPUS:0030290782

SN - 0378-4754

VL - 42

SP - 595

EP - 605

JO - Mathematics and Computers in Simulation

JF - Mathematics and Computers in Simulation

IS - 4-6

ER -