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**Discrete boundary element methods on general meshes in 3D.**
*(English)*
Zbl 0966.65090

The authors study stability and convergence of fully discrete Galerkin methods for boundary integral equations on bounded piecewise smooth surfaces in \(\mathbb R^3.\) The integral operators have to be strongly elliptic, but the meshes are not assumed to be quasiuniform. The methods are treated by generalisations of standard inverse estimates to the non-quasiuniform case.

The authors analyse the integration errors for nearly singular and smooth Galerkin integrals in order to determine the accuracy of quadrature rules ensuring that the resulting discrete Galerkin method enjoys the same stability and convergence properties as the true Galerkin method.

The theory, which covers quite general surfaces and meshes, is applied to linear boundary elements on triangles and corresponding quadrature rules. As an example the required accuracy of quadrature rules for the standard integral operators from Laplace’s equation is given which preserve the stability and convergence properties of Galerkin’s method in the energy norm.

The authors analyse the integration errors for nearly singular and smooth Galerkin integrals in order to determine the accuracy of quadrature rules ensuring that the resulting discrete Galerkin method enjoys the same stability and convergence properties as the true Galerkin method.

The theory, which covers quite general surfaces and meshes, is applied to linear boundary elements on triangles and corresponding quadrature rules. As an example the required accuracy of quadrature rules for the standard integral operators from Laplace’s equation is given which preserve the stability and convergence properties of Galerkin’s method in the energy norm.

Reviewer: Gunther Schmidt (Berlin)

### MSC:

65N38 | Boundary element methods for boundary value problems involving PDEs |

65N12 | Stability and convergence of numerical methods for boundary value problems involving PDEs |

35J05 | Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation |