Trefftz method: A general theory

Flexible approximation schemes with numerical and semi‐analytical bases

“…Another approach that has already been tried before is adaptive FLAME that locally reduces the grid size, thereby extending the effectiveness of relatively simple analytical bases (Dai and Tsukerman, 2008). FLAME is connected to several other classes of numerical methods such as variational Trefftz (Jirousek and Zielinski, 1997;Herrera, 2000), generalized FEM (Melenk and Babuška, 1996 and others), discontinuous Galerkin (Cockburn et al, 2000;Arnold et al, 2002 and many others), variational-difference schemes of Moskow et al (1999), discontinuous enrichment and finite increment calculus (Hughes et al, 1998;Farhat et al, 2001;Oñate, 2000). These connections were previously discussed in (Tsukerman 2006) (see in particular Figure 1, Sections 1 and 2, and references there) and are not revisited here to save space.…”

Flexible approximation schemes with numerical and semi‐analytical bases

“…Another approach that has already been tried before is adaptive FLAME that locally reduces the grid size, thereby extending the effectiveness of relatively simple analytical bases (Dai and Tsukerman, 2008). FLAME is connected to several other classes of numerical methods such as variational Trefftz (Jirousek and Zielinski, 1997;Herrera, 2000), generalized FEM (Melenk and Babuška, 1996 and others), discontinuous Galerkin (Cockburn et al, 2000;Arnold et al, 2002 and many others), variational-difference schemes of Moskow et al (1999), discontinuous enrichment and finite increment calculus (Hughes et al, 1998;Farhat et al, 2001;Oñate, 2000). These connections were previously discussed in (Tsukerman 2006) (see in particular Figure 1, Sections 1 and 2, and references there) and are not revisited here to save space.…”

A Method of Fundamental Solutions in Poroelasticity to Model the Stress Field in Geothermal Reservoirs

“…Here, we assumed Dirichlet boundary data given on the whole boundary . For a more general approach, see, e.g., [129]. An overview on the Trefftz method can be found, e.g., in [158].…”

A direct method of parameter estimation for steady state flow in heterogeneous aquifers with unknown boundary conditions

“…A similar concept is the Method of Fundamental Solution [ Kupradze and Aleksidze , 1964], which is mathematically equivalent to the Trefftz method as the number of coefficients of the individual superimposed functions increases to infinity [ Li et al , 2010]. Application of the Trefftz method was initially focused on inverse problems, before the current focus on BVP, with both finite element formulations [ Jirousek and Zielinski , 1997; Kita and Kamiya , 1995; Herrera , 2000; Li et al , 2008; Kolodziej and Zielinski , 2009] and meshless method [ Galybin and Mukhamediev , 2004].…”

Flexible approximation schemes with numerical and semi‐analytical bases

“…Another approach that has already been tried before is adaptive FLAME that locally reduces the grid size, thereby extending the effectiveness of relatively simple analytical bases (Dai and Tsukerman, 2008). FLAME is connected to several other classes of numerical methods such as variational Trefftz (Jirousek and Zielinski, 1997;Herrera, 2000), generalized FEM (Melenk and Babuška, 1996 and others), discontinuous Galerkin (Cockburn et al, 2000;Arnold et al, 2002 and many others), variational-difference schemes of Moskow et al (1999), discontinuous enrichment and finite increment calculus (Hughes et al, 1998;Farhat et al, 2001;Oñate, 2000). These connections were previously discussed in (Tsukerman 2006) (see in particular Figure 1, Sections 1 and 2, and references there) and are not revisited here to save space.…”

A Method of Fundamental Solutions in Poroelasticity to Model the Stress Field in Geothermal Reservoirs

“…Here, we assumed Dirichlet boundary data given on the whole boundary . For a more general approach, see, e.g., [129]. An overview on the Trefftz method can be found, e.g., in [158].…”

A direct method of parameter estimation for steady state flow in heterogeneous aquifers with unknown boundary conditions

“…A similar concept is the Method of Fundamental Solution [ Kupradze and Aleksidze , 1964], which is mathematically equivalent to the Trefftz method as the number of coefficients of the individual superimposed functions increases to infinity [ Li et al , 2010]. Application of the Trefftz method was initially focused on inverse problems, before the current focus on BVP, with both finite element formulations [ Jirousek and Zielinski , 1997; Kita and Kamiya , 1995; Herrera , 2000; Li et al , 2008; Kolodziej and Zielinski , 2009] and meshless method [ Galybin and Mukhamediev , 2004].…”

Flexible approximation schemes with numerical and semi‐analytical bases

“…Another approach that has already been tried before is adaptive FLAME that locally reduces the grid size, thereby extending the effectiveness of relatively simple analytical bases (Dai and Tsukerman, 2008). FLAME is connected to several other classes of numerical methods such as variational Trefftz (Jirousek and Zielinski, 1997;Herrera, 2000), generalized FEM (Melenk and Babuška, 1996 and others), discontinuous Galerkin (Cockburn et al, 2000;Arnold et al, 2002 and many others), variational-difference schemes of Moskow et al (1999), discontinuous enrichment and finite increment calculus (Hughes et al, 1998;Farhat et al, 2001;Oñate, 2000). These connections were previously discussed in (Tsukerman 2006) (see in particular Figure 1, Sections 1 and 2, and references there) and are not revisited here to save space.…”

A Method of Fundamental Solutions in Poroelasticity to Model the Stress Field in Geothermal Reservoirs

“…Here, we assumed Dirichlet boundary data given on the whole boundary . For a more general approach, see, e.g., [129]. An overview on the Trefftz method can be found, e.g., in [158].…”

A direct method of parameter estimation for steady state flow in heterogeneous aquifers with unknown boundary conditions

“…A similar concept is the Method of Fundamental Solution [ Kupradze and Aleksidze , 1964], which is mathematically equivalent to the Trefftz method as the number of coefficients of the individual superimposed functions increases to infinity [ Li et al , 2010]. Application of the Trefftz method was initially focused on inverse problems, before the current focus on BVP, with both finite element formulations [ Jirousek and Zielinski , 1997; Kita and Kamiya , 1995; Herrera , 2000; Li et al , 2008; Kolodziej and Zielinski , 2009] and meshless method [ Galybin and Mukhamediev , 2004].…”