Trefftz‐like numerical method for linear boundary‐value problems

Abstract A generalized Trefftz method for solution of linear boundary‐value problems for partial differential equation sets is presented. An approximate solution is constructed as a linear combination of basis functions satisfying the differential equations involved and forming a complete sequence. Unknown factors are determined through minimizing some norm of the approximate solution boundary‐value deviation from the current boundary data. A unified formulation constitutes the foundation for various modifications and particular applications. The approach enables one to: reduce the problem geometrical dimensionality by one; construct a smooth approximate solution satisfying the governing differential equation; use a universal set of approximating functions for problems with identical equations and varying in boundary conditions; develop a closed, semi‐analytical computational algorithm requiring small input and producing extremely high output accuracy; obtain approximate solution error bounds in inner points via its deviation from prescribed values on the boundary and easily control the degree of accuracy of the numerical results by the residual values at boundary points; treat solution singularities caused by singular boundary conditions precisely. The technique developed has proved to be effective in fluid mechanics, elastostatics, Saint‐Venant's theory for torsion and bending prisms, potential and elastic plate bending problems.